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regernachweis ergibt Hyemophilus influencae. J G Vorwiegend allergisches Asthma bronchiale. J14 G Pneumonie durch Haemophilus influenzae. B! ICD J Vorwiegend allergisches Asthma bronchiale Allergische: Bronchitis o.n.A. Allergische: Rhinopathie mit Asthma bronchiale Atopisches Asthma. Asthma bronchiale, allergisches; extrinsisches Asthma, atopisches Asthma, exogenes allergisches Asthma bronchiale. ICD Code J Definition Asthma. F54, J, Psychogenes Asthma bronchiale. J, Vorwiegend allergisches Asthma bronchiale. Allergische Bronchitis. Allergische Rhinopathie mit Asthma. Diagnosen: JG Allergisches Bronchialasthma J A Ausgeschlossen: Lungenemphysem JV Verdacht auf Allergische Rhinopathie durch Pollen.

ICD J - Vorwiegend allergisches Asthma bronchiale. Medikamente und Infos zu ICD Code J ICD J - Asthma bronchiale, nicht näher bezeichnet. Medikamente und Infos zu ICD Code J ICD J Vorwiegend allergisches Asthma bronchiale Allergische: Bronchitis o.n.A. Allergische: Rhinopathie mit Asthma bronchiale Atopisches Asthma. Some further information on equations of article source case will be found in 8. If criticising Beste Spielothek in HСЊttenheim finden opinion J45.0g unequal, integration gives an inverse trigonometric or hyperbolic func- tion; see any table of integrals, that of Peirce and Foster, for example. The first, which contains a term in ifl, is an extension of the Riccati equation ; see Constants of integration. I believe, however, that it will help people who want to solve a differential equation. An infinite series; see Al Try the following cases. The Linear Equation 13 3.

J45.0g Präparate der ICD-10 Gruppe J45.9 - Asthma bronchiale, nicht näher bezeichnet

Während die Asthma-Prävalenz in den vergangenen Jahrzehnten in vielen Ländern zugenommen hat, scheint die Progression in westlichen Ländern zum Stillstand gekommen zu sein. Plana u. In: Bitcoin GebГјhren of Allergy and Clinical Immunology. Band 6, Nummer 1, FebruarS. Beeh, F. Leibniz-Sozietät der Wissenschaften zu Berlin, J45, Asthma bronchiale. J, Vorwiegend allergisches Asthma bronchiale. J​1, Nichtallergisches Asthma bronchiale. J, Mischformen des Asthma. Der ICD Code J45 beschreibt Chronische Krankheiten der unteren Atemwege (​JJ47), konkret Asthma bronchiale. Mischformen des Asthma bronchiale (Kombination aus J und J) Asthma J G Pneumonie durch Klebsiella pneumoniae. EIGENE NOTIZEN. ICD J - Asthma bronchiale, nicht näher bezeichnet. Medikamente und Infos zu ICD Code J ICD J - Vorwiegend allergisches Asthma bronchiale. Medikamente und Infos zu ICD Code J

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Die indische Regierung unterstützt die jährliche Zeremonie der Familie Bathini Goud in Hyderabad , an der rund Menschen teilnehmen, mit Sonderzügen. In: Journal of Allergy and Clinical Immunology. Auflage, 6. In: BMC Medicine. Prävalenzdaten zu ärztlich diagnostiziertem Asthma sind häufig inhomogen. Namensräume Artikel Diskussion. Samuel Samson — nutzte check this out Therapie die Lobeliewofür er das erste Patent erhielt, die Wurzel der Lobelia Syphyliticadie in die Pharmacopoeia of Massachussett's Medical Society aufgenommen wurde, und die Tinktur der Blätter, die in die US-Pharmakopöe einging. Bitte hierzu den Hinweis zu Gesundheitsthemen beachten! J45.0g, W.

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No Missing Variables 61 3. Homogeneous Equations and Related Types 63 4. The Method of Lagrange 66 6. Change of Variable 66 7.

The General Equation 67 8. Singular Solutions 74 The Linear Equation of Second Order 82 1. Equations Reducible to Type 1 86 4.

Transformations of the Linear Equation 87 5. The Exact Equation 92 6. Resolution of Operators - 94 7. Solution of the Equation in Infinite Form 95 8.

Series Solution 96 9. Equations of Standard Type Integral Solution Continued Fraction Solution The Linear Nonhomogeneous Equation B2.

The Nonlinear Equation of Second Order 1. The Dependent Variable Is Missing 2. The Independent Variable Is Missing 3.

Homogeneous Equations 4. The Exact Nonlinear Equation 5. Change of Variable 6. The Equation with Fixed Critical Points 8.

Painlev6 Equations Singular Solutions C. Transformations of the Equation 5. The Exact Equation 6. Series Solution 7. Integral Solutions 9.

The Linear Nonhomogeneous Equation C2. The Nonlinear Equation of Second Order 5. Two alternative procedures are offered.

Go to Part I. It contains most of the methods which have been developed for solving ordinary differential equations.

If directions are followed, it should be possible to classify the given equation as a special case of one or more general types.

When the type has been identified, details of the method for solving it will also be found.

Go to Part II. It contains more than two thousand ordinary diff- erential equations, arranged in a systematic order. A solution of each is also given or some directions as to a method.

References to the general methods of Part I are included in each case. If the equation to be solved is contained in this collection, a solution of it is thus found.

If the equa- tion is not in the collection, one of similar form there may suggest a method that might be tried. Otherwise, it will be necessary to consult Part I and seek a general method which is applicable.

The remainder of this section contains some definitions, some general properties of differential equations, and a description of the symbols used in the book.

Some Definitions. A differential equation is a relation involving one or more derivatives and an unknovni function.

The problem of solving it is a search for that unknown function. The solution of a diff- erential equation is any relation, free from derivatives, which satisfies the equation identically.

It will often also contain parameters or literal constants. None of these cases are treated in this book. The order of an ordinary differential equation is the order of the highest differential coefficient which it contains.

It is often necessary to rationalize the equation and to clear it of fractions in order to determine the degree.

In some cases, a transcendental function of one or more deri- vatives may occur. Typical examples are In y', sin y", cosh In such cases, the equation is of infinite degree.

The Solution of a Differential Equation. Given a function F x,y,Ci,C2, A total of n differential equations could be found in this way, because any one of the constants Ci could be selected for elimination.

Continue in this way and use 2, 3, Eaeh ooatdns n—k constants mid there are i-l-1 independent equarions. The reverse of this is also true.

Given a diffiwential equation of order n, the existence of a function 2 , containing n arbitrary constants, is implied. Such a function is the complete primitive of 1 , its general or complete solution, its general or complete integral, or its integral curves.

Some writers use the word integral to designate an implicit relation like 2 and restrict the meaning of solution to an explicit relation for y as a function of x.

Others use integral or solution to mean the result obtained when 1 is' solved, reserving the word primitive for 2 , if the differential equation has resulted from it by differentiation.

Such distinctions will not be made here and the words primitive, integral, solution will mean any function that satisfies the differential equation identically.

The n arbitrary constants eliminated from 2 in obtaining 1 , or resulting from the latter when it is solved, are the constants of integration.

It should not be inferred, however,. Nevertheless, if the general solution can be found, the differential equation could have resulted from it by the process described.

In many cases, especially in applied mathematics, it is of interest to find solutions which satisfy certain special conditions required for physical reasons.

Such conditions are initial values or boundary conditions. The required solution can usually be obtained by assigning particular values to one or more of the integration constants in the general solution.

The result, containing less than the full number of arbitrary constants, still satisfies the differential equation.

It is a special or particular solution or special or particular integral. Sometimes a function exists which satisfies the differential equation but which is not a special base of the general solution.

It is a singular solution or a singular integral. Existence Theorems. Statements in the preceding section were based on the assumption that the given differential equation actually has a unique solution.

Elementary methods for finding it are successful beoause both the equation and its solution belong to a relatively simple Aass of functions.

The details and proofs of them are lengthy; they will not be given here. For suitable references see Ince- 1 , Coddington Mid Levinson, or other sources listed in the Bibliography.

For our purposes, they can be briefly summarized in terms of a real variable. A given function of the variables x, y, 2 , Since, within certain limits, these n constants are arbitrary, they are the required integration constants of the general solution.

Further comments about existence theorems will be made later in connection with special kinds of differential equations.

The symbols x, y are used for the independent and dependent variables respectively in this book.

Other variables that may occur are v, w, etc. They will be defined when they are used. The arbitrary constant of integration is C or Ci, C 2 , The letters m, n mean integers in most cases.

Greek letters have generally been avoided but they have been used where necessary. Primes, double primes, etc. A wide variety of symbols is needed in a book such as this and it is not always easy to be consistent.

The general principles just stated wdll be used, as far as possible. Departures from them will be explained as they occur.

A number in parentheses always refers to an equation. If there are subsections, like , , , etc.

Section numbers are indicated in a similar way but without parentheses. Thus, the number appearing in any part of A1 would mean section If it referred to section of A2, it would be given as A When further subdivisions of a section are required they are called a, b, c, etc.

If still another subdivision is wanted, Roman numerals i, ii, ill, are used. References to texts or other books are given with the name or names of the authors.

Full details, such as the title of the book, the publisher, date of publication, etc. An equation of first order B. An equation of second order C.

An equation of order higher than two A. Proceed according to one of the following directions. The equation is a pol3momial in y'. The degree of the polynomial is the degree of the differential equation.

There are two oases. The degree of the equation is unity; see Al. The degree of the equation is two or greater; see A2.

The equation is not a pol3momial in y'. If the equation is algebraic in y' but not a polynomial or if transcendental functions of y' occur, like cos y', In y', etc.

The form is also useful when two new variables are to be taken. Compare the given equation with the cases of the following sections.

As far as practicable, they have been presented in the order of increasing complexity. Often an equation may be solved by more than one method but it will usually be true that the bne with the smaller type number will be simpler.

Thus, the problem has become one of integral calculus, not one of difieren- tial equations. The other will usually also disappear, leaving a relation between Ci and Cz.

For a special case, see Dependent Variable Missing. Neither Variable Missing. Independent Variable Missing. The Euler Equation.

The equation is separable, see , but evaluation of the integrals may cause difficulty; see c and d.

Square the equation to get one of the second degree and try the methods of A2, which may be successful when some of the Of vanish.

In most oases, the procedure of c or d will be preferable. If the roots of the quadratic are equal, integration is simple. If they are unequal, integration gives an inverse trigonometric or hyperbolic func- tion; see any table of integrals, that of Peirce and Foster, for example.

Such functions can be inverted to give an algebraic solution, but see also d. The functions X x and Y y are of third or fourth degree.

The resulting integrals define elliptic functions and their properties can be used to secure an algebraic solution of the differential equation.

Con- venient references for elliptic integrals and functions are Erd4l3d-1, Macrobert, Whittaker and Watson. After some tedious algebra, both can be rationalized.

The method of Cauchy also called the method of Euler. Assume a scfiution like 1 , where the ct are constants to be determined.

This means that five Ci can be fixed, the sixth one remains arbitrary, and that an infinite number of polynomials like 1 will satisfy the differential equation.

The ct in 1 could be determined by equating coefficients of equal powers of the two variables. The calculations can be simplified by geometric considerations, as shown by Jacobi, and the general integral of the differential equation can be presented as a fourth-order determinant, as shown by Stieljes.

The details are given by Goursat. Obviously, the coefficients in 2 are also satisfactory. The general integral 1 is a family of curves of the fourth degree, with two double points at infinity on the x and y axes, respectively.

The family of corves has an envelope composed of eight straight lines, four each parallel to the X and y axes. These are singular solutions of the differential equation; see A2.

See The solution is d. A special solution, yi is known. Two special solutions, yi and yz, are known. Three special solutions are known.

The general case, where neither f x nor g x vanish. The solution previously given can be found in several different ways.

The method of Lagrange or variation of parameter. The general solution follows. See also Bl The method of Bernoulli.

Determine v x so that the term in parentheses vanishes. Iliis is case b. The general solution follows when u and v are substituted into y uv.

The integrating factor. The linear equation may be converted into an exact equation, see 7, by means of an integrating factor. The Riccati Equation There are three different types but the one actually studied by Riccati is a special case, cf.

If the coefficients in these equations have certain special proper- ties, they can be solved by quadratures.

In general, the solutions define functions more complicated than the elementary transcendental func- tions ; see also B Such functions are logarithmic, exponential, trigo- nometric, or hyperbolic; and simple combinations of these functions.

They are obtained by integration of algebraic functions or by rational processes following such an integration.

The Riccati equation is introduced at this point because its form sug- gests that it is similar to the linear equation, type 2.

Actually, it is a more difficult equation to solve and it, together with type 4 following. Is much more difficult than many of the subsequent cases; see also The Generalized Riccati Equation.

It is a special case of the Abel equation; see and There are several special cases. It is linear; see 2. The equation is of Bernoulli type; see 5.

Other special cases. Compare the given equation with the types of and If it is one of these, proceed as directed there. The general equation.

None of the preceding special cases are applic- able. Refer to , in the hope of transforming the equation to types or If this is not successful, go to The General Solution of the Equation.

If one or more special solutions of 1 can be found by inspection or otherwise, the general solution is easy to obtain.

When a special solution is not obvious, go to , where some methods for finding one are presented. A qseoial solution is known.

The general solution can be found by quadratures in two different ways. The general solution is found with only one quadrature.

Four special equations are known. Transform the equation into one of second order. It is linear and of second order; see Bl.

Special Solutions of the Riccati Equation. Sometimes one or more special equations can be found by inspection or by a lucky guess.

More formal procedures apply in certain cases. If the present section is not helpful, try le or some transformation in There are two possibilities.

The degree of F x is odd. There is no polynomial solution of 3 ; hence none of 1. The degree of F x is even. Two possible pol3momial solutions may exist.

The coefficients in the polynomial could be found by a simple modification of the square-root extraction method of elementary algebra, by the method of undetermined coefficients, or by expansion in a Maolaurin series.

K neither is a special solution of 3 , there are no polynomial solutions of 3 and hence none of 1.

There is no polynomial solution of either 2 or 1. Alternatively, it is possible to state conditions under which both polynomials are solu- tions of the differential equation ; see c.

This is the necessary and sufficient condition that both solutions of b satisfy the differential equation. There are two cases.

The general solution of 1 follows. Polynomial coefficients. This property restricts the possible types of polynomial solutions.

See e for further discussion of this case. Several polynomial solutions. Given 4 , any specified number of polynomial solutions can be constructed.

The considerations of this section may not be too useful in solving a Biccati equation. They could, however, be very helpful in constructing an equation with a predetermined 'number of polynomial solutions.

Some Properties of the Rlccati Equation. The following properties of 1 sometimes apply in attempting to solve it. Removal of the linear term.

The result, which is similar to 3 , can be achieved in three different ways. It might then be treated as in , but see also'b for a- special case.

Relatioiis between the ooefBcients. When certain relations exist between the coefficients of 1 its solution may be easy to obtain.

Use a to remove the linear term in 1. Then, if F x is propor- tional to G z in 5 , the result is separable, type See also The Riccati Equation.

The constant m need not be an integer; a and c are constants. Test the equation for integrability in finite form by a. If the test is successful, go to b.

Sometimes, it is helpful to transform the equation into one of second order; see c. Integrable oases. For other values of k, the equation is not integrable in finite terms; return to The sign in 8 is positive.

This equation is 8 with a negative sign. Gro to case ii. The sign in 8 is negative, or case 1 has been transformed into this type.

The equation has been trans- formed back again into case 1, with k decreased by unity. Successive applications of the transformations in 1 and 11 will eventually reduce k to zero and 7 will then be separable, type The general solution in such integrable cases is best completed by the method of A transformed equation.

Given 7 to solve in an integrable case, transform it to 9 and solve it for w x , according to This is generally easier than the successive reductions described in a.

A second-order equation. The result is often expressed in terms of Bessel functions. A Special, Rlccatl Equation.

Under cer- tain conditions, the equation can be integrated in finite form; see a. For the relation between 10 and the Biccati equation of , see b.

Examine the equation to see if it satisfies one of the following conditions. If it does not, return to Conversion to type Given an equation of that type to solve, con- vert it to the form of 10 and follow the directions in a.

Such a procedure is generally simpler than the direct solution of type The Abel Equation There are two equations of this type.

The first, which contains a term in ifl, is an extension of the Riccati equation ; see An equation of the second kind is a further generalization; see In either case, finite solutions result only when the equation has certain special properties.

Some of these cases are described in the two following sections. In the general case, it may be necessary to use one of the procedures given in Abel Equation of the First Kind.

The equation is separable, type The equation is linear, type 2. A Riccati equation results ; see 3. The equation is of Bernoulli type ; see 5.

Reduction to standard form. This can be chosen as the can- onical or standard form of the Abel equation. See also g.

The equation can be converted into one of the second order. Separable equations. There are two cases where a separable equa- tion can be produced by suitable transformation of variables.

In both cases an elliptic integral occurs. Abel Equation of the Second Kind. Try one of the following procedures in the general case.

The result is an Abel equation of the first kind ; see The coefficients are rather complicated functions of fi, gt, and the derivatives of gf.

It is assumed that 0 and that go, gi are differentiable. These effects are caused by the variable transformation and 8 is not misprinted.

The equation can be converted into one of the first kind, see , or into a simpler equation of the second kind. It can be simplified further; see e.

This equation is separable, type Restore y after it has been solved and the result is equivalent to 2. Equations Linear in the Variabies There are two different types.

For some other equations of similar form, see There are three possibilities ; a more general equation is given in It is separable, type There are three possible cases.

The equation which results is separable, type To test a differential equation for exactness, see , where methods for finding the primitive are also given.

If the test for exactness fails, it means that common factors have been removed and an integrating factor is needed.

Such factors can alwajrs be found, in principle, but not always easily in practice. When they are known or can be found, the equation can be made exact and integrated; see The Exact Equation.

The equation is exact and also homogeneous, with degree k — 1. Go to 8, for the solution can be found without quadrature.

The equation is exact but not homogeneous. The solution can be found in two equivalent ways. The quantities xq, yo in the integrals are arbitrary.

They are most conveniently chosen to make the integrations easy, frequently as 0 or 1. The equation is not exact. Proceed to in the hope of finding some function that will make 3 exact so that the methods of this section can be used.

The Integrating Factor. A general theory, based on the properties of continuous transformation groups, shows that an integrating factor can be found, at least in principle, for a properly classified equation.

The solution of the equation can then be completed by quadratures. The use of group theory for finding the integrating factor will not be described here.

For some appropriate references see Cohen, Ince-1, Lie, Page. Since the integrating factor may not always be easy to find, some other procedure may be simpler.

One possibility is a suitable variable trans- formation, converting the given differential equation to one of the types Iff previously considered.

See also 8 and 9 for further suggestions. If the methods of this section are preferred, try to , in turn.

The number of integrating factors. Two integrating factors. An equation for the integrating factor. Unfortunately, the partial differential equation may not be easy to solve and, if one must use it to find an integrating factor, another method might be preferred for solving the given ordinary differential equation.

However, the general solution of the partial differential equation is not needed; any special solution of it will suffice.

In some cases, the partial differential equation becomes an ordinary differential equation and then it may be easy to find an integrating factor.

Some examples are given in The equation is homogeneous. The integration may be completed without a quadrature ; see 8. Integrating Factor by Inspection.

In some cases, an inte- grating factor is obvious from the form of the given differential equation or it is obtainable after a few lucky guesses.

In making such guesses, the following properties may be helpful. If more formal procedures are preferred, go at once to If possible, separate the equation into two or more parts, one of which is exact and the others inexact.

It is only necessary to find an integrating fgictor for the single inexact part. Then, convert F x, y to a function of u.

Examine the differential equation to see if it contains terms like those in Table 1. In more complicated cases, use the methods of a mr b.

Integrating Factors for Special Equations. It is assumed that an integrating factor was not apparent by inspection, as suggested in , and that more formal procedures are wanted.

When the equation is of some special type, the integrating factor can be given at once, as shown in this section. Alternatively, certain tests can be made on the equation and, if they hold, the integrating factor follows.

For this case, see To some extent, the two procedures duplicate each other. Proceed with this section or go directly to , as desired.

There are a number of special cases. Both M and N cannot vanish. If either vanishes identically, the reciprocal of the other is an integrating factor.

TABLE 1. In this case, see 8, the integrating factor is IjM. The equation is isobaric. This case is a generalization of d.

The exponents need not be integers. Here u, v are any functions of x and y. Thus, this case is more general than d. The equation is linear; see 2.

Exchange of variables. Tests for an Integrating Factor. A quick test will then show whether or not such an integrating factor exists.

To proceed in this way. The success of the method depends on the proper choice of u x, y. A prominent combination of terms in the differential equation might suggest something to be tried.

See also Table 1. The integrating factor depends on only one variable. If the result is that shown in the first column of Table 2, the integrating factor is that given in the second column.

The letters a and k are constants. For a more general case, see b. TABLX 2. When u - yjx, the condition can be generalized; see c.

The integrating factor, see Table 3, is a homogeneous function of x and y of degree zero ; see 8. However, in the exceptional case, the inte- grating factor is unnecessary since the variables are separable.

Advantage can be taken of the fact that k is unspecified for a homogeneous equation. Select two different values of k 0 and 1 might be suitable and find two different integrating factors.

The general solution of the equation follows from b. When the differential equation itself is not homogeneous, the method of this section will work if 7 is satisfied for some value of k.

Homogeneous Equations and Related Types The word homogeneous is used with two different meanings in the study of differential equations.

For the second meaning of homogeneous, see Bl. The test for homoge- neity can usually be made by inspection. In that case, the methods of , , or may be used.

If the equation is homogeneous, but not exact, and an integrating factor is not to be sought, see Some related equations are presented in ff.

The Homogeneous Equation. The given equation has the form 2. The nota- tion means that x is to be replaced in both P and Q by unity and that y is to be replaced by u.

This solution might be contained in 5 or it might be a singular solution; see A Li some cases, polar coordinates are useful.

The Isobaric Equation. It is not necesaaiy that n be an integer. Itisnotneoes- sacy that n be an integer. Take x as the dependent variable and u as the independent variable.

The Jacobi Equation. A generalization of it is given in For a special case, see a. Otherwise, there are three different ways of solving 6 ; see b, c, d.

A special case. The general case. Finally, take X as the dependent variable and a Bern- oulli equation is obtained; see 5.

An alternative method. The Darboux equation, see , is a generali- zation of the Jacobi equation. Methods of solving the former can also be applied to the latter.

Equation numbers here refer to those shown in All Ki are different. A third method. This one, quite different from b or c, depends on solving three simultaneous equations with constant coefficients.

Such equations are not treated in this book but the particular one needed here is quite simple. Details can be found in a number of places, for example, Ince-1, Kaplan- 1.

Its characteristic roots are Ki and its arbitrary constants are Ci. Thus, the solution of the simultaneous system of equations will also give the solution of the Jacobi equation.

There are three cases. When the variable t is eliminated, they will correspond to the three cases of c.

All are different. The Darboux Equation. This is a generalization of , where the fi x, y are polynomials in x and y of maximum degree m and at least one of them is actually of degree m.

The required conditions in the general case were obtained by Darboux. The details may be found in Ince-1 or Goursat. Here, we only state the two condi- tions which lead to the complete solution of the Darboux equation.

The Ki can be found from 8 and the general solution of the differential equation re- sults without quadratures.

Two oases arise. The determinant of the coefficients of the pi vanishes. The case has become equivalent to a. The determinant of the coefficients does not vanish.

Change of Variable When an equation does not fit into one of the previous types, it can often be made to do so by a suitable transformation of variables.

In fact, some of the preceding methods were based on a variable change. It was stated in a that if an equation has a unique solution it will have an infinite number of integrating factors.

Similarly, if it has a unique solu- tion it can be solved by a change of variable. In practice, it may not al- ways be easy to find either an integrating factor or the proper new vari- able.

Sometimes, one method is preferred; at othw times, the seomd method is successful. The following sections contain a few sqggestions that might be helpful if one or two new variables are sought.

Many examples of tiik type will be found in Part II. New Independent Variable. Interchange x and y. The former becomes the new dependent variable and the latter, the new independent variable.

This is an especially simple trick for it requires little calculation and it is easy to see whether the equation is so converted into a known type.

Two examples, where this method was successful, were given; see h and New Dependent Variable. No general rules can be given, but a conspicuous function in the given equation is often suggestive.

Three rather general equations of no previous type, where this procedure works, are given in b, c, d. Some hints are presented in a. It is not necessary that m and n be integers in this section.

The new variable is u x. Try the following cases. Other- wise, it might be chosen as some function of x which appears in the original equation.

See also 2, , , , , , , , , , , and Part II. In the first case, try or trigonometric functions like sftiay, cos ay, tan ay, etc.

For examples, see , 5, 6, and Part II. This equation is reminiscent of several types from other sections. Whoi fix a 0, a Bernoulli equation results; see 5.

None ofthe preceding special oases occur. This equation, like that of b, is similar to, but not identical with, several previous types.

There are several cases. It is a Biccati equation; see It is an Abel equation; see Two New Variables. The Legendre Transformation. Also take a new dependent variable, Y — xp —y.

The method of this section is often applicable to equations of degree two or higher; see A2. The solutions would then all be simple combinations of algebraic or elementary transcendental functions; see 3.

Mathematicians, however, have preferred another procedure and have sought algebraic differential equations which define new transcen- dental functions through their solutions.

In doing so, they have studied oertaiR classes of differential equations which will be presented in the apjffopriate places later in this book.

Solutions as an infinite series; see Solution by an approximate method; see In the usual case, a will be preferred. The remaining parts Of tiiis section summarize some general properties of 1.

They are inserted here for convenience of reference and can be consulted as needed. Analytic Functions. It is the purpose of this section to consider some properties of analytic functions as used in the study of differential equa- tions.

For further details and proofs of the statements which follow, see appropriate references in the Bibliography, for example, Ince-1, Codding- ton and Levinson.

The test for analyticity can be be made simply, for it is only necessary, according to 3 , that the function and its derivatives exist at the point in question.

There are several kinds of singular points. Such a singularity is called a poh of order n or a nonessential singular point.

The misbehavior of the function is effectively avoided by use of the multi- plicative term. If a circle can be drawn with center at the singular point so that no other singular point is enclosed, the singular point is an isolated one.

Within a circle of radius less than 2ln, there are an infinite number of poles. Essential singular points. It is said to be an essential singular point.

Branch points. Whatever the order, some derivatives of finite order and all higher ones will be infinite at a branch point.

Fixed and movable singular points. Examination of the coefficients in a differential equation will reveal the nature of its singular points, which can be of the kinds described in a, b, c.

These are the fixed or intrinsic singular points of the differential equation. It does not follow that solu- tions of the equation will also have singular points of the same kind.

Only under special conditions is the origin a singular point of the solution. Such a singular point, which moves about as the initial values are varied, is a movcMe or parametric singular point.

Singular points for linear equations are always fixed. A nonlinear equation of first order and of first degree can have movable poles and movable branch points but no movable essential singularities.

Nonlinear equations of second or higher order can have movable singular points of all kinds. Movable branch points and essential singularities, excluding poles of finite order, are often called critical points of the differential equation.

There are several cases, depending on the nature of fix, y. Proceed as foUows. Since the latter constant is arbitrary it will be the constant of integration.

For some comments on convergence, see d. The method of undetermined coefficients. Replace the left-hand side of the differential equation by the derivative of 2.

The result, which no longer contains y, is an identity in x. Equate coefficients of equal powers of x in this equation and obtain relations with which Ai, Az,.

There are three possibilities. All coefficients after At are zero so Uiat 2 is a polynomial of degree k.

The present procedure can, of course, be used in such cases, if desired. A general law results for the coefficients in terms of Ao.

Such a relation is called a two-term recursion formula or a, first-order difference equation; see also iii. There is no general law for the coefficients so that three-term for- mulas or even more complicated ones occur.

This means that At will usually depend on At-z, etc. In fact, it may depend in some rather involved way on all of the coefficients which precede it and no explicit solution can be found for A as a function of Aq.

It will become more and more laborious to calculate successive coefficients but, nevertheless, such calculations may be continued as long as desired.

Many-term recursion formulas are linear finite-difference equations. It is often convenient to study them by such methods; see, for example, Jordan, Milne- 1, MOne- Thompson.

Frequently, one wishes a series solution so that y can be determined within some specified limits of error. This will fix the number of coeffi- cients which must be calculated; see also d.

It should be noted that the existence theorems guarantee that the solution is valid but it is not possible to make tests for convergence, as in ii, since the general term is unknown.

Expansion in a Taylor series. The method is equivalent to that of a, which in the usual case will be easier to apply. Use this result, together with and fy, to calculate y" in 3.

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The details and proofs of them are lengthy; they will not be given here. For suitable references see Ince- 1 , Coddington Mid Levinson, or other sources listed in the Bibliography.

For our purposes, they can be briefly summarized in terms of a real variable. A given function of the variables x, y, 2 , Since, within certain limits, these n constants are arbitrary, they are the required integration constants of the general solution.

Further comments about existence theorems will be made later in connection with special kinds of differential equations.

The symbols x, y are used for the independent and dependent variables respectively in this book. Other variables that may occur are v, w, etc.

They will be defined when they are used. The arbitrary constant of integration is C or Ci, C 2 , The letters m, n mean integers in most cases.

Greek letters have generally been avoided but they have been used where necessary. Primes, double primes, etc. A wide variety of symbols is needed in a book such as this and it is not always easy to be consistent.

The general principles just stated wdll be used, as far as possible. Departures from them will be explained as they occur.

A number in parentheses always refers to an equation. If there are subsections, like , , , etc. Section numbers are indicated in a similar way but without parentheses.

Thus, the number appearing in any part of A1 would mean section If it referred to section of A2, it would be given as A When further subdivisions of a section are required they are called a, b, c, etc.

If still another subdivision is wanted, Roman numerals i, ii, ill, are used. References to texts or other books are given with the name or names of the authors.

Full details, such as the title of the book, the publisher, date of publication, etc. An equation of first order B.

An equation of second order C. An equation of order higher than two A. Proceed according to one of the following directions. The equation is a pol3momial in y'.

The degree of the polynomial is the degree of the differential equation. There are two oases. The degree of the equation is unity; see Al.

The degree of the equation is two or greater; see A2. The equation is not a pol3momial in y'. If the equation is algebraic in y' but not a polynomial or if transcendental functions of y' occur, like cos y', In y', etc.

The form is also useful when two new variables are to be taken. Compare the given equation with the cases of the following sections. As far as practicable, they have been presented in the order of increasing complexity.

Often an equation may be solved by more than one method but it will usually be true that the bne with the smaller type number will be simpler.

Thus, the problem has become one of integral calculus, not one of difieren- tial equations. The other will usually also disappear, leaving a relation between Ci and Cz.

For a special case, see Dependent Variable Missing. Neither Variable Missing. Independent Variable Missing.

The Euler Equation. The equation is separable, see , but evaluation of the integrals may cause difficulty; see c and d. Square the equation to get one of the second degree and try the methods of A2, which may be successful when some of the Of vanish.

In most oases, the procedure of c or d will be preferable. If the roots of the quadratic are equal, integration is simple.

If they are unequal, integration gives an inverse trigonometric or hyperbolic func- tion; see any table of integrals, that of Peirce and Foster, for example.

Such functions can be inverted to give an algebraic solution, but see also d. The functions X x and Y y are of third or fourth degree.

The resulting integrals define elliptic functions and their properties can be used to secure an algebraic solution of the differential equation.

Con- venient references for elliptic integrals and functions are Erd4l3d-1, Macrobert, Whittaker and Watson. After some tedious algebra, both can be rationalized.

The method of Cauchy also called the method of Euler. Assume a scfiution like 1 , where the ct are constants to be determined.

This means that five Ci can be fixed, the sixth one remains arbitrary, and that an infinite number of polynomials like 1 will satisfy the differential equation.

The ct in 1 could be determined by equating coefficients of equal powers of the two variables. The calculations can be simplified by geometric considerations, as shown by Jacobi, and the general integral of the differential equation can be presented as a fourth-order determinant, as shown by Stieljes.

The details are given by Goursat. Obviously, the coefficients in 2 are also satisfactory. The general integral 1 is a family of curves of the fourth degree, with two double points at infinity on the x and y axes, respectively.

The family of corves has an envelope composed of eight straight lines, four each parallel to the X and y axes. These are singular solutions of the differential equation; see A2.

See The solution is d. A special solution, yi is known. Two special solutions, yi and yz, are known.

Three special solutions are known. The general case, where neither f x nor g x vanish. The solution previously given can be found in several different ways.

The method of Lagrange or variation of parameter. The general solution follows. See also Bl The method of Bernoulli. Determine v x so that the term in parentheses vanishes.

Iliis is case b. The general solution follows when u and v are substituted into y uv. The integrating factor. The linear equation may be converted into an exact equation, see 7, by means of an integrating factor.

The Riccati Equation There are three different types but the one actually studied by Riccati is a special case, cf.

If the coefficients in these equations have certain special proper- ties, they can be solved by quadratures. In general, the solutions define functions more complicated than the elementary transcendental func- tions ; see also B Such functions are logarithmic, exponential, trigo- nometric, or hyperbolic; and simple combinations of these functions.

They are obtained by integration of algebraic functions or by rational processes following such an integration.

The Riccati equation is introduced at this point because its form sug- gests that it is similar to the linear equation, type 2.

Actually, it is a more difficult equation to solve and it, together with type 4 following. Is much more difficult than many of the subsequent cases; see also The Generalized Riccati Equation.

It is a special case of the Abel equation; see and There are several special cases. It is linear; see 2. The equation is of Bernoulli type; see 5.

Other special cases. Compare the given equation with the types of and If it is one of these, proceed as directed there. The general equation.

None of the preceding special cases are applic- able. Refer to , in the hope of transforming the equation to types or If this is not successful, go to The General Solution of the Equation.

If one or more special solutions of 1 can be found by inspection or otherwise, the general solution is easy to obtain.

When a special solution is not obvious, go to , where some methods for finding one are presented. A qseoial solution is known.

The general solution can be found by quadratures in two different ways. The general solution is found with only one quadrature. Four special equations are known.

Transform the equation into one of second order. It is linear and of second order; see Bl. Special Solutions of the Riccati Equation.

Sometimes one or more special equations can be found by inspection or by a lucky guess. More formal procedures apply in certain cases.

If the present section is not helpful, try le or some transformation in There are two possibilities. The degree of F x is odd.

There is no polynomial solution of 3 ; hence none of 1. The degree of F x is even. Two possible pol3momial solutions may exist. The coefficients in the polynomial could be found by a simple modification of the square-root extraction method of elementary algebra, by the method of undetermined coefficients, or by expansion in a Maolaurin series.

K neither is a special solution of 3 , there are no polynomial solutions of 3 and hence none of 1. There is no polynomial solution of either 2 or 1.

Alternatively, it is possible to state conditions under which both polynomials are solu- tions of the differential equation ; see c.

This is the necessary and sufficient condition that both solutions of b satisfy the differential equation. There are two cases.

The general solution of 1 follows. Polynomial coefficients. This property restricts the possible types of polynomial solutions.

See e for further discussion of this case. Several polynomial solutions. Given 4 , any specified number of polynomial solutions can be constructed.

The considerations of this section may not be too useful in solving a Biccati equation. They could, however, be very helpful in constructing an equation with a predetermined 'number of polynomial solutions.

Some Properties of the Rlccati Equation. The following properties of 1 sometimes apply in attempting to solve it. Removal of the linear term.

The result, which is similar to 3 , can be achieved in three different ways. It might then be treated as in , but see also'b for a- special case.

Relatioiis between the ooefBcients. When certain relations exist between the coefficients of 1 its solution may be easy to obtain.

Use a to remove the linear term in 1. Then, if F x is propor- tional to G z in 5 , the result is separable, type See also The Riccati Equation.

The constant m need not be an integer; a and c are constants. Test the equation for integrability in finite form by a.

If the test is successful, go to b. Sometimes, it is helpful to transform the equation into one of second order; see c. Integrable oases.

For other values of k, the equation is not integrable in finite terms; return to The sign in 8 is positive. This equation is 8 with a negative sign.

Gro to case ii. The sign in 8 is negative, or case 1 has been transformed into this type. The equation has been trans- formed back again into case 1, with k decreased by unity.

Successive applications of the transformations in 1 and 11 will eventually reduce k to zero and 7 will then be separable, type The general solution in such integrable cases is best completed by the method of A transformed equation.

Given 7 to solve in an integrable case, transform it to 9 and solve it for w x , according to This is generally easier than the successive reductions described in a.

A second-order equation. The result is often expressed in terms of Bessel functions. A Special, Rlccatl Equation.

Under cer- tain conditions, the equation can be integrated in finite form; see a. For the relation between 10 and the Biccati equation of , see b.

Examine the equation to see if it satisfies one of the following conditions. If it does not, return to Conversion to type Given an equation of that type to solve, con- vert it to the form of 10 and follow the directions in a.

Such a procedure is generally simpler than the direct solution of type The Abel Equation There are two equations of this type.

The first, which contains a term in ifl, is an extension of the Riccati equation ; see An equation of the second kind is a further generalization; see In either case, finite solutions result only when the equation has certain special properties.

Some of these cases are described in the two following sections. In the general case, it may be necessary to use one of the procedures given in Abel Equation of the First Kind.

The equation is separable, type The equation is linear, type 2. A Riccati equation results ; see 3. The equation is of Bernoulli type ; see 5.

Reduction to standard form. This can be chosen as the can- onical or standard form of the Abel equation. See also g. The equation can be converted into one of the second order.

Separable equations. There are two cases where a separable equa- tion can be produced by suitable transformation of variables. In both cases an elliptic integral occurs.

Abel Equation of the Second Kind. Try one of the following procedures in the general case. The result is an Abel equation of the first kind ; see The coefficients are rather complicated functions of fi, gt, and the derivatives of gf.

It is assumed that 0 and that go, gi are differentiable. These effects are caused by the variable transformation and 8 is not misprinted.

The equation can be converted into one of the first kind, see , or into a simpler equation of the second kind. It can be simplified further; see e.

This equation is separable, type Restore y after it has been solved and the result is equivalent to 2. Equations Linear in the Variabies There are two different types.

For some other equations of similar form, see There are three possibilities ; a more general equation is given in It is separable, type There are three possible cases.

The equation which results is separable, type To test a differential equation for exactness, see , where methods for finding the primitive are also given.

If the test for exactness fails, it means that common factors have been removed and an integrating factor is needed.

Such factors can alwajrs be found, in principle, but not always easily in practice. When they are known or can be found, the equation can be made exact and integrated; see The Exact Equation.

The equation is exact and also homogeneous, with degree k — 1. Go to 8, for the solution can be found without quadrature. The equation is exact but not homogeneous.

The solution can be found in two equivalent ways. The quantities xq, yo in the integrals are arbitrary. They are most conveniently chosen to make the integrations easy, frequently as 0 or 1.

The equation is not exact. Proceed to in the hope of finding some function that will make 3 exact so that the methods of this section can be used.

The Integrating Factor. A general theory, based on the properties of continuous transformation groups, shows that an integrating factor can be found, at least in principle, for a properly classified equation.

The solution of the equation can then be completed by quadratures. The use of group theory for finding the integrating factor will not be described here.

For some appropriate references see Cohen, Ince-1, Lie, Page. Since the integrating factor may not always be easy to find, some other procedure may be simpler.

One possibility is a suitable variable trans- formation, converting the given differential equation to one of the types Iff previously considered.

See also 8 and 9 for further suggestions. If the methods of this section are preferred, try to , in turn. The number of integrating factors.

Two integrating factors. An equation for the integrating factor. Unfortunately, the partial differential equation may not be easy to solve and, if one must use it to find an integrating factor, another method might be preferred for solving the given ordinary differential equation.

However, the general solution of the partial differential equation is not needed; any special solution of it will suffice.

In some cases, the partial differential equation becomes an ordinary differential equation and then it may be easy to find an integrating factor.

Some examples are given in The equation is homogeneous. The integration may be completed without a quadrature ; see 8.

Integrating Factor by Inspection. In some cases, an inte- grating factor is obvious from the form of the given differential equation or it is obtainable after a few lucky guesses.

In making such guesses, the following properties may be helpful. If more formal procedures are preferred, go at once to If possible, separate the equation into two or more parts, one of which is exact and the others inexact.

It is only necessary to find an integrating fgictor for the single inexact part. Then, convert F x, y to a function of u. Examine the differential equation to see if it contains terms like those in Table 1.

In more complicated cases, use the methods of a mr b. Integrating Factors for Special Equations. It is assumed that an integrating factor was not apparent by inspection, as suggested in , and that more formal procedures are wanted.

When the equation is of some special type, the integrating factor can be given at once, as shown in this section.

Alternatively, certain tests can be made on the equation and, if they hold, the integrating factor follows. For this case, see To some extent, the two procedures duplicate each other.

Proceed with this section or go directly to , as desired. There are a number of special cases.

Both M and N cannot vanish. If either vanishes identically, the reciprocal of the other is an integrating factor.

TABLE 1. In this case, see 8, the integrating factor is IjM. The equation is isobaric. This case is a generalization of d.

The exponents need not be integers. Here u, v are any functions of x and y. Thus, this case is more general than d.

The equation is linear; see 2. Exchange of variables. Tests for an Integrating Factor. A quick test will then show whether or not such an integrating factor exists.

To proceed in this way. The success of the method depends on the proper choice of u x, y. A prominent combination of terms in the differential equation might suggest something to be tried.

See also Table 1. The integrating factor depends on only one variable. If the result is that shown in the first column of Table 2, the integrating factor is that given in the second column.

The letters a and k are constants. For a more general case, see b. TABLX 2. When u - yjx, the condition can be generalized; see c.

The integrating factor, see Table 3, is a homogeneous function of x and y of degree zero ; see 8. However, in the exceptional case, the inte- grating factor is unnecessary since the variables are separable.

Advantage can be taken of the fact that k is unspecified for a homogeneous equation. Select two different values of k 0 and 1 might be suitable and find two different integrating factors.

The general solution of the equation follows from b. When the differential equation itself is not homogeneous, the method of this section will work if 7 is satisfied for some value of k.

Homogeneous Equations and Related Types The word homogeneous is used with two different meanings in the study of differential equations. For the second meaning of homogeneous, see Bl.

The test for homoge- neity can usually be made by inspection. In that case, the methods of , , or may be used.

If the equation is homogeneous, but not exact, and an integrating factor is not to be sought, see Some related equations are presented in ff.

The Homogeneous Equation. The given equation has the form 2. The nota- tion means that x is to be replaced in both P and Q by unity and that y is to be replaced by u.

This solution might be contained in 5 or it might be a singular solution; see A Li some cases, polar coordinates are useful. The Isobaric Equation.

It is not necesaaiy that n be an integer. Itisnotneoes- sacy that n be an integer. Take x as the dependent variable and u as the independent variable.

The Jacobi Equation. A generalization of it is given in For a special case, see a. Otherwise, there are three different ways of solving 6 ; see b, c, d.

A special case. The general case. Finally, take X as the dependent variable and a Bern- oulli equation is obtained; see 5.

An alternative method. The Darboux equation, see , is a generali- zation of the Jacobi equation. Methods of solving the former can also be applied to the latter.

Equation numbers here refer to those shown in All Ki are different. A third method. This one, quite different from b or c, depends on solving three simultaneous equations with constant coefficients.

Such equations are not treated in this book but the particular one needed here is quite simple.

Details can be found in a number of places, for example, Ince-1, Kaplan- 1. Its characteristic roots are Ki and its arbitrary constants are Ci.

Thus, the solution of the simultaneous system of equations will also give the solution of the Jacobi equation. There are three cases.

When the variable t is eliminated, they will correspond to the three cases of c. All are different. The Darboux Equation.

This is a generalization of , where the fi x, y are polynomials in x and y of maximum degree m and at least one of them is actually of degree m.

The required conditions in the general case were obtained by Darboux. The details may be found in Ince-1 or Goursat. Here, we only state the two condi- tions which lead to the complete solution of the Darboux equation.

The Ki can be found from 8 and the general solution of the differential equation re- sults without quadratures.

Two oases arise. The determinant of the coefficients of the pi vanishes. The case has become equivalent to a.

The determinant of the coefficients does not vanish. Change of Variable When an equation does not fit into one of the previous types, it can often be made to do so by a suitable transformation of variables.

In fact, some of the preceding methods were based on a variable change. It was stated in a that if an equation has a unique solution it will have an infinite number of integrating factors.

Similarly, if it has a unique solu- tion it can be solved by a change of variable. In practice, it may not al- ways be easy to find either an integrating factor or the proper new vari- able.

Sometimes, one method is preferred; at othw times, the seomd method is successful. The following sections contain a few sqggestions that might be helpful if one or two new variables are sought.

Many examples of tiik type will be found in Part II. New Independent Variable. Interchange x and y. The former becomes the new dependent variable and the latter, the new independent variable.

This is an especially simple trick for it requires little calculation and it is easy to see whether the equation is so converted into a known type.

Two examples, where this method was successful, were given; see h and New Dependent Variable. No general rules can be given, but a conspicuous function in the given equation is often suggestive.

Three rather general equations of no previous type, where this procedure works, are given in b, c, d. Some hints are presented in a.

It is not necessary that m and n be integers in this section. The new variable is u x. Try the following cases. Other- wise, it might be chosen as some function of x which appears in the original equation.

See also 2, , , , , , , , , , , and Part II. In the first case, try or trigonometric functions like sftiay, cos ay, tan ay, etc. For examples, see , 5, 6, and Part II.

This equation is reminiscent of several types from other sections. Whoi fix a 0, a Bernoulli equation results; see 5. None ofthe preceding special oases occur.

This equation, like that of b, is similar to, but not identical with, several previous types. There are several cases. It is a Biccati equation; see It is an Abel equation; see Two New Variables.

The Legendre Transformation. Also take a new dependent variable, Y — xp —y. The method of this section is often applicable to equations of degree two or higher; see A2.

The solutions would then all be simple combinations of algebraic or elementary transcendental functions; see 3.

Mathematicians, however, have preferred another procedure and have sought algebraic differential equations which define new transcen- dental functions through their solutions.

In doing so, they have studied oertaiR classes of differential equations which will be presented in the apjffopriate places later in this book.

Solutions as an infinite series; see Solution by an approximate method; see In the usual case, a will be preferred.

The remaining parts Of tiiis section summarize some general properties of 1. They are inserted here for convenience of reference and can be consulted as needed.

Analytic Functions. It is the purpose of this section to consider some properties of analytic functions as used in the study of differential equa- tions.

For further details and proofs of the statements which follow, see appropriate references in the Bibliography, for example, Ince-1, Codding- ton and Levinson.

The test for analyticity can be be made simply, for it is only necessary, according to 3 , that the function and its derivatives exist at the point in question.

There are several kinds of singular points. Such a singularity is called a poh of order n or a nonessential singular point.

The misbehavior of the function is effectively avoided by use of the multi- plicative term. If a circle can be drawn with center at the singular point so that no other singular point is enclosed, the singular point is an isolated one.

Within a circle of radius less than 2ln, there are an infinite number of poles. Essential singular points.

It is said to be an essential singular point. Branch points. Whatever the order, some derivatives of finite order and all higher ones will be infinite at a branch point.

Fixed and movable singular points. Examination of the coefficients in a differential equation will reveal the nature of its singular points, which can be of the kinds described in a, b, c.

These are the fixed or intrinsic singular points of the differential equation. It does not follow that solu- tions of the equation will also have singular points of the same kind.

Only under special conditions is the origin a singular point of the solution. Such a singular point, which moves about as the initial values are varied, is a movcMe or parametric singular point.

Singular points for linear equations are always fixed. A nonlinear equation of first order and of first degree can have movable poles and movable branch points but no movable essential singularities.

Nonlinear equations of second or higher order can have movable singular points of all kinds. Movable branch points and essential singularities, excluding poles of finite order, are often called critical points of the differential equation.

There are several cases, depending on the nature of fix, y. Proceed as foUows. Since the latter constant is arbitrary it will be the constant of integration.

For some comments on convergence, see d. The method of undetermined coefficients. Replace the left-hand side of the differential equation by the derivative of 2.

The result, which no longer contains y, is an identity in x. Equate coefficients of equal powers of x in this equation and obtain relations with which Ai, Az,.

There are three possibilities. All coefficients after At are zero so Uiat 2 is a polynomial of degree k. The present procedure can, of course, be used in such cases, if desired.

A general law results for the coefficients in terms of Ao. Such a relation is called a two-term recursion formula or a, first-order difference equation; see also iii.

There is no general law for the coefficients so that three-term for- mulas or even more complicated ones occur.

This means that At will usually depend on At-z, etc. In fact, it may depend in some rather involved way on all of the coefficients which precede it and no explicit solution can be found for A as a function of Aq.

It will become more and more laborious to calculate successive coefficients but, nevertheless, such calculations may be continued as long as desired.

Many-term recursion formulas are linear finite-difference equations. It is often convenient to study them by such methods; see, for example, Jordan, Milne- 1, MOne- Thompson.

Frequently, one wishes a series solution so that y can be determined within some specified limits of error. This will fix the number of coeffi- cients which must be calculated; see also d.

It should be noted that the existence theorems guarantee that the solution is valid but it is not possible to make tests for convergence, as in ii, since the general term is unknown.

Expansion in a Taylor series. The method is equivalent to that of a, which in the usual case will be easier to apply.

Use this result, together with and fy, to calculate y" in 3. Ciontinue in this way to find the third derivative and as many more terms as may be wanted.

Evaluation of the successive derivatives can become quite complicated, if it is necessary to use more than a few terms in the series ; see also d.

The coefficients by integration. For an alternative way of deter- mining the coefficients in the series 2 , repeated integrations may be used rather than differentiations.

For the details, see Convergence of the series solution. If the conditions required by the existence theorem hold, it is certain that 2 is the general solution of 1.

A more exact statement of these conditions may be useful. If further details and proofs are wanted, other sources must be consulted.

Some suitable references are Coddington and Levinson, Goursat, Ince Take Jf equal to or greater than this sum, so that! If t is fixed, the right-hand side of 4 will decrease as x decreases.

It is still possible, in this case, to solve the differential equation. All coefficients vanish at X 0, which means that each term in the right-hand member of the differential equation contains x as a factor.

The Equation of Briot and Bouquet. This is case a of For a generalization of it, see There are three possible situations.

Successive values of the At are obtained from 7. Nonanalytic solutions may also exist ; see c. There are relations between k and the coefficients atj so that 7 can be used to calculate the coefficients At.

The preceding result can be obtained in another way. The last differential equatipa is similar to 5 but the coefficient of u has been reduced by unity.

There are no solutions analytic at the origin but there is a general solution con- taining an arbitrary constant.

It is a series in x and x In x; it approaches zero as x approaches zero along a properly chosen path. For references and further details, see Ince Nonanalytic solutions.

The first coefficient cqo is arbitrary. The series approaches zero as x approaches zero along a properly chosen path.

More complete details are given by Goursat, Ince-1, Valiron. V The Generalized Equation of Briot and Bouquet. It is mMiiiTiftd that both P and Q are convergent double series in x and y, aimiUr to the right-hand member of 5.

However, P and Q are not divisible by any power of x or y. They are described, for example, by Ince 1.

Uo is a simple root. The equation is now of type and can be handled further as described there, b.

Uo is a double root. There are two conclusions, both for Jb 0. The origin is an essential singular point; see lO-I. There are two special cases of interest, which in terms of the original variables, are as follows.

The second equation is of Ricoati type; see Thus 2 is an integral equation which should be solved. Continue in this way and calculate The work is stopped with a solution which is a sufficiently close approxi- mation to the exact solution.

The procedure may become quite tedious, since the integrations are usually more and more difficult to perform.

With appropriate restrictions, existence theorems show that the successive integrals converge and that the solution is unique.

It may then be possible to find an approximate solution by one of the following means: a. For these reasons, we limit the descrip- tion of them here.

Graphical methods. Considered geometrically, the differential equation assigns a slope to every point x, y in the x, y-plane. Each will determine a small portion of an integral curve.

The general shape of that curve will become apparent as the plotting is continued. A smooth curve with the line segments as tangents will be the integral curve sought.

This situation is equivalent to the general solution if xq is a constant and yo is a variable parameter or the constant of integration.

Following these general principles, many ingenious variations have been devised. As one possibility, isoclines are first drawn.

These are the loci along which the slope required by the differential equation has a constant value.

When a number of isoclines have been drawn, the integral curves can then be sketched on the same plot.

As another possibility, a first approximation to an integral curve could be drawn and this could be improved by redrawing, until two successive curves coincide.

To compute correction. In other cases, a nomograph might be constructed for calculating the corrections. No high order of precision can be expected by gra]thical means but the results may be acceptable for special problems, especially if the graphs are carefully constructed and of sufficiently large size.

Lacking some par- ticular reason for using this method, one of those in c would usually be preferred. For more details about the graphical methods, the book of Levy and Baggott is recommended.

Mechanical methods. Devices for mechanical integration have a long history. Two such instruments are the integraph and the polar planimeter.

The former, described by Abdank-Abakanowicz in , contains a tracing point, which movps along a graph of the integrand. An attached pen draws the integral curve.

The polar planimeter, one model of which was invented by Amsler in , also has a point, which moves along the integrand curve.

A coimected scale and vernier indicates the number of complete or fractional revolutions made in tracing out the perimeter of the integrand curve.

A simple calibration and calculation give the area under 'this curve, which, of course, equals the value of the definite integral. Either of these devices could be used to solve a simple differential equation in several ways.

For example, the integral equation method, see 12, would give successive approximations to the solution, when the various integrals had been evaluated mechanically.

A wide variety of more sophisticated instruments has been described in the literature and many of them actually constructed.

In some oases, the machine has been invented to solve a particular type of equation, such as that of Bicoati, see 3, or of Abel, see 4.

In other cases, the machine may be more versatile but none of them are simple and all are expensive to build. For many references and further description of such instru- ments, see Kamke.

It was based on addition and integra- tion. The former was achieved by gear boxes and the latter, by a wheel and disk mechanism, similar to that of the polar planimeter.

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