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Solving Second-Order Differential Equations by Decomposition

By Fritz Schwarz

Submitted: September 24th 2020Reviewed: November 11th 2020Published: January 4th 2021

DOI: 10.5772/intechopen.94993

Downloaded: 67

Abstract

The subject of this article are linear and quasilinear differential equations of second order that may be decomposed into a first-order component with guaranteed solution procedure for obtaining closed-form solutions. These are homogeneous or inhomogeneous linear components, special Riccati components, Bernoulli, Clairaut or d’Alembert components. Procedures are described how they may be determined and how solutions of the originally given second order equation may be obtained from them. This makes it possible to solve new classes of differential equations and opens up a new area of research. Applying decomposition to linear inhomogeneous equations a simple procedure for determining a special solution follows. It is not based on the method of variation of constants of Lagrange, and consequently does not require the knowledge of a fundamental system. Algorithms based on these results are implemented in the computer algebra system ALLTYPES which is available on the website www.alltypes.de.

Keywords

  • ordinary differential equations
  • decomposition
  • exact solutions
  • computer algebra

1. Introduction

The history of differential equations begins shortly after the establishment of the analysis by Newton and Leibniz in the 17th century. A brief overview of its first hundred years can be found in Appendix A of Ince’s book [1]. These early investigations were mainly limited to first-order equations, associated with the names Riccati, Bernoulli and Euler. Starting in the early 18th century special linear equations of higher order were also investigated.

A more systematic search for solution methods was initiated by the results of Galois for solving algebraic equations in the early 19th century. Inspired by these results, Picard and Pessiot in Paris founded a solution theory for linear differential equations, known as Picard-Vessiot theory or differential Galois theory. A good introduction into their work and its extensions by Loewy may be found in the books [2, 3]. Completely independent of these activities Sophus Lie in Leipzig founded the so-called symmetry analysis for solving nonlinear differential Equations [4, 5]. Its main weaknesses are that most differential equations have no symmetries and therefore it cannot be applied. Furthermore, there are many differential equations with fairly simple closed form solutions that have no symmetries. That was essentially the status in the early twentieth century, which did not fundamentally change until its end.

In this situation, a new solution method based on decompositions was proposed [6]. Essentially a decomposition means to find a component of lower order such that the original equation may be represented as a differential polynomial in terms of this component. Its existence is based on the following observation. Let Fxyyy=0be a second-order differential equation for a function ydepending on a variable x, and ωxyC1C2=0its general solution depending on two undetermined constants. It describes a two-parameter family of curves in the xy-plane. If C1and C2are constrained by a relation φC1C2=0the resulting expression for ωcontains effectively a single parameter C. It describes a family of curves that may obey a first-order differential equation called a component. Its solutions are also solutions of the originally given second-order equation.

Every second-order equation has an infinite number of first-order components corresponding to the choice of φC1C2. Any such component has the form

Fxyyy=fxyzzCzgxyyC.E1

Its meaning may be described as follows. If zgxyyCis substituted into fxyzzCthe second-order equation on the left-hand side is obtained. The constant Cdoes not necessarily occur in fand g, the same is true for yand its occurence in f.

Solving a second-order equation by decomposition involves two steps. First a decomposition of a certain type has to be found. Then the first order equation has to be solved in order to get the solutions of the original second-order equation. Of particular interest are those components the solution of which can always be determined. These are linear homogeneous and inhomogeneous components, special Riccati components, Bernoulli, Clairaut or d’Alembert components.

In this article equations of second order for an unknown function ydepending on xwith leading term yor yyare considered. They are assumed to be linear in y, polynomial in the derivatives y, and rational in yand x. Equations of this kind are fairly common in applications, therefore many special examples of them are given in the collections by Kamke [7], Murphy [8], Polyanin [9], Sachdev [10] and Zwillinger [11]. Many interesting applications of such differential equations can be found in the textbooks by MacCluer et al. [12] and Swift and Wirkus [13].

In the following Section 2 equations with leading term yare considered, and possible linear or Bernoulli components are determined. For linear inhomogeneous equations it is shown how decomposition leads to a new procedure for determining a special solution without first having to know a fundamental system. Equations with leading term yyand possible components of Clairaut or d’Alembert type are the subject of Section 3. Most of the examples do not have Lie symmetries, so decomposition is the only way to solve them. The last Section 4 discusses various possible generalizations of the decomposition method, on the one hand more general equations to be solved, on the other hand more general first-order components.

2. Equations with leading term y

Equations that are linear in the highest derivative y, but may contain powers of ywith coefficients that are rational in yand xare considered in this section. Moreover it is assumed that they are primitive, i.e. the leading coefficient is unity. Their general form is

y+k=0Kckxyyk=0withckxyQxy,KN.E2

Equations of this form appear in numerous applications, as can be seen in the collections of solved examples quoted above. The following proposition has been proved in [6], it is the basis for generating quasilinear first-order components; as usual ydydxand Dddx.

Proposition 1 Let a second-order quasilinear Eq. (2) be given. A first-order component zy+rxyexists if rxysatisfies

rxrryk=0K1kckxyrk=0.E3

Then the original second-order equation can be decomposed as

zryz+k=1Kckxyzrk1krkzy+r=0.E4

The proof may be found in Section 2 of [6]. As a first application linear first-order components of the form zy+axy+bxare searched for, i.e. with the above notation rxy=axy+bx; its coefficients aand bare solutions of the so-called determining system, they may be in any field extension of Qx. The following proposition describes how they may be obtained.

Proposition 2. Let a second-order quasilinear Eq. (2) be given. In order that it has a linear first-order component zy+axy+bxthe coefficients axand bxhave to satisfy

aa2y+babk=0K1kckxyay+bk=0.E5

Then (2) may be written as follows

zaz+k=1Kckxyzaybk1kay+bkzy+ay+b=0.E6

The coefficients aand bare solutions of a first-order algebro-differential system. Its general form is

aa2+pabx=0,  bab+qabx=0,  riabx=0E7

for i=1,2; pabx, qabxand riabxare polynomials in aand b, and rational in x; their maximal degree in aand bis K. The riabxgenerate an ideal IabQxab.

Proof. Substituting r=axy+bxinto (3) yields (5). At this point yis considered as an undetermined function. Therefore the left-hand side of (5) is represented as a partial fraction in y. Equating its coefficients to zero yields sufficient conditions in order that (5) vanishes and zis a component of (2). The first order ode’s for aand bin the determining system (7) originate from the coefficients of first and zeroth degree in yof (5). The polynomials in aand b, i.e. pabx, qabxand riabxin (7) originate from the powers of ay+band the rational coefficients ckxyin (5), i.e. exclusively from the nonlinearities of (2). Substitution of y=zaybinto (4) yields (6). As a result, the sums at the left-hand side of (6) are a polynomial in zthe coefficients of which may depend explicitly on y.

It is important to represent the left side of (5) as a partial fraction in y, only in this way the structure of the system (7) is assured.

2.1 Linear equations

If K=1, c1xy=c1xand c0xy=c0xy+crxthe above proposition contains the decomposition of linear equations as a special case as shown next.

Corollary 1 Let K=1, c1xy=c1x, c0xy=c0xy+crxand the linear inhomogeneous second-order equation

y+c1xy+c0xy+crx=0E8

be given. A first-order component zy+axy+bxexists if aand bare solutions of the determining system

aa2+c1xac0x=0  and  b+c1xabcrx=0.E9

If it is satisfied Eq. (8) may be written as

z+c1azzy+ay+b=0.E10

Proof. The system (9) follows from (5) for the given special values of Kand the coefficients ck. Then reduction of (8) w.r.t. zyields (10).

It is remarkable that in the case of linear equations the algebraic conditions riabxare missing, i.e. they are the most significant contributions originating from possible nonlinearities in (2).

For linear homogeneous ode’s, i.e. for cr=0and b=0, Loewy decompositions have been shown to be an effective method for determining a fundamental system [3]. It is based on a factorization of the linear differential operator corresponding to the given equation over its base field, i.e. restricting the coefficients of the factors to the field of the coefficients of the given second-order equation. This restriction does not apply in the above corollary, the coefficients may be in any field extension.

For linear inhomogeneous equations in addition to a fundamental system a special solution has to be found. The above corollary avoids the usual method of variation of constants that somehow appears like an ad hoc method. The method described in the above corollary requires only a special solution of a Riccati equation and subsequently solving a linear first-order equation in order to obtain the general solution of the second-order Eq. (8). The following example applies this procedure.

Example 1 The equation

yy1xy=x+1exE11

is Equation 2.109 in Kamke’s collection [7]. Here c1=1, c0=1xand cr=x+1ex. The Riccati equation aa2a+1x=0has the special solution a=11x. From b+1xb=x+1exfollows b=Cx+1xx2x+1exand leads to the component

z=y1+1xy+1xC+x2x+1ex.

Integration yields the general solution

y=C1xex+C2xexexpxdxx2+x2xlogx1ex.

This is also the general solution of Eq. (11).

It may occur that a fundamental system of a second-order equation is rather complicated. Usually this is the case when the Riccati equation for ain (9) does not have a special rational solution and the usual algorithms for solving it do not apply, but one of the special cases of Section 4.9 (a), ... (e) in [7]. Then it may be advantageous to assume that all integration constants in (9) are zero and only a special solution is determined as shown next.

Example 2 Consider the equation

y12xy+xy+1=0.E12

Here c1=12x, c0=xand cr=1. The Riccati equation aa212xax=0has the special solution a=xtan23xx, it yields

b12x+xtan23xb1=0.

Its special solution leads to the component

zy+xtan23xxy+xcos23xxcos23xxdxx.

One more integration yields a special solution of (12).

y0=cos23xxxcos23xx2cos23xxdxxdx.

The application of Corollary 1 is particularly convenient if the coefficients c1and c0are constant and the solutions of the algebraic equation a2c1a+c0=0are also solutions of the Riccati equation for a. The following example is of this type.

Example 3 The equation y+4y+4y=coshxhas coefficients c1=c0=4and cr=coshx. The solution of a22=0is a=2. It leads to b+2b=coshxand the component

z=y+2y+Cexp2x+13sinhx23sinhx.

Its general solution

y=C1exp2x+C2xexp2x+59coshx49sinhx

is also the general solution of the given second-order equation.□

2.2 Quasilinear equations

The most interesting applications of Proposition 2 relate to nonlinear equations, of course. They differ from the linear case mainly by the occurence of the ideal Iabin (7), which defines algebraic conditions riabx=0for the coefficients of a possible component. Furthermore, the first-order ode’s for aand bare modified due to the nonlinearity by additional terms. The structure of the determining system (7) suggests the following solution procedure.

At first the algebraic system riabx=0is established and a Gröbner basis for the ideal Iabis generated. Usually it may be determined rather efficiently.

If it is inconsistent a linear component does not exist in any field extension. This applies to a generic nonlinear equation of the form (2).

If the ideal Iabis finite-dimensional each solution that satisfies the two first-order ode’s yields a component that may be integrated and leads to a one-parameter family of solutions of the given second-order equation.

Finally, the algebraic equations may generate a relation between aand b; substitution into the first-order differential equations may lead to one of the above cases, or to a solution depending on a parameter. In the latter case a one-parameter family of linear components exists, integrating the corresponding equation yields the general solution of the given second-order equation containing two undetermined constants.

Subsequently this proceeding will be illustrated by several examples. They show that all of the alternatives mentioned actually exist.

Example 4 Consider the equation

y+xy2+x1yy+xx+1yy21x+1y=0.

Its coefficients c2=x, c1=x1y+xx+1and c0=y2yx+1result in the system

aa22abx+axx+1+bx1=0,  babb2x+bxx+1+1x+1=0,a2x1xa1x=0.

The single algebraic equation has the solutions a=1xand a=1and the decompositions

z+xz2+x3y+2x2y+x2+xy+x+1x2+xzzy1xy=0,z+xz2x2y+2xy+y+1x+1zzy+y=0

follow. Integration of the two components yields the two one-parameter families y=Cexpxand y=Cxof solution curves. It is not obvious how the general solution of the second-order equation involving two constants is supported by them. □

The most interesting, of course, are equations that allow a one-parameter family of linear components and whose integration gives the general solution. The next example is of this type.

Example 5 Consider the equation

yyy2+23y1xy2=0E13

with K=2and the coefficients c2=1y, c1=23yand c0=yx; they generate the determining system

a+1x=0,  b+ab+23a=0,  b2+23b=0

with solution a=logx+C, b=23from which the decomposition

z1yz21ylogxyCy+23zzylogxCy23=0

is obtained. Integration of the first-order component leads to the general solution

y=23xxexpC1xexpC1xdxxx+C2

of Eq. (13), it does not have a Lie symmetry.

Here the question arises how exceptional are the equations that have a one-parameter family of linear components of the first order and thus have a general solution in closed form. The following example is a generalization of the previous one. A family of second order equations is constructed whose general solution can be given explicitly.

Example 6 The equation

yyy2+pxy+qxy2=0E14

with undetermined coefficients pxand qxgeneralizes the preceding example. Here c2=1y, c1=pxyand c0=qxy. A first-order linear component zy+ay+bexists if aand bare solutions of the system

aqx=0,  b+ab+pxa=0,  bb+px=0.

The result may be described as follows. If px=kis a constant, and qxis an undetermined function then a=qxdx+C, b=kand the decomposition

y1yy2+kyy+qxy=z1yz2+qxdx+Czkyzzy+qxdx+Cyk=0

exists. Defining QxC1qxdx+C1integration of the first-order component yields

y=expQxC1dxkexpQ(xC1dx)+C2.

This is the general solution of Eq. (14). □

It turns out that a behavior similar to that in the previous example often applies, i.e. first-order linear components often exist not only for isolated equations, but for entire families, which are parameterized by indefinite functions. This explains the existence of families of solvable equations as those given in the collections mentioned above.

Bernoulli equations are another class of first-order ode’s with guaranteed closed form general solutions. In addition to a term linear in ythey contain a nonlinearity ynwhere nis an integer; n=1or n=0correspond to linear homogeneous or linear inhomogeneous equations, respectively. Similar as for linear components, a special Bernoulli component guarantees a one-parameter set of solution curves of a given second-order equation, and a one-parameter family of such components guarantees the general solution of the latter. The main result of this section is the following proposition.

Proposition 3 Let a second-order quasilinear Eq. (2) be given. In order that it has a first-order Bernoulli component zy+axyn+bxy, nN, the coefficients aand bhave to satisfy

an+1abyn+bb2yna2y2n1k=0K1kckxyayn+byk=0.E15

Then (2) may be written as follows

znayn1+bz+k=1Kzaynbyk+1k+1ayn+bykzy+ayn+by=0.E16

The coefficients aand bmay be obtained from a first-order algebro-differential system; its general form is

an+1ab+pabx=0,bb2+qabx=0,riabx=0;E17

pabx, qabxand riabxare polynomials in aand b, and rational in x; the maximal degree in aand bis K, they generate an ideal Iabin the ring Qxab.

Proof. Substituting r=axyn+bxyinto (3) yields condition (15). Representing its left-hand side as partial fraction in the variable y, the coefficients of the various terms yield sufficient conditions for its vanishing. They form the algebro-differential system (17). The first order ode’s for aand boriginate from the coefficients of nth and first degree in y, respectively; pabx, qabxand riabxoriginate from the coefficients ckxyand the powers of ayn+by.

Substitution of y=zaynbyinto (4) yields (16). As a result, the sums at the left-hand side of (16) are a polynomial in zand zthe coefficients of which may depend explicitly on y. □

The structure of the system (17) is similar as for linear components considered above, and consequently also the proceeding for its solution. The following examples applies the above proposition.

Example 7 The equation

yyy2+2y3y+xy2=0E18

with K=2has coefficients c2=1y, c1=2y2and c0=xy. For generic nthe condition

an1abyn+bxyn1a2y2n1+2ayn+2+2by3=0E19

follows. The two equations bx=0and b=0originating from the coefficients of yand y3, respectively, are inconsistent. In order for a Bernoulli component to exist, this inconsistency must be compensated by other coefficients for a suitable choice of n. To this end either n=1or n=3is required. The former leads to the inconsistency x=0, whereas the latter yields

a2ab+2b=0,bx=0,a2a=0.

This system has the solution a=1, b=12x2+Cfrom which the decomposition

z1yz2+y2+12x2+Czzy+y3+12x2+Cy=0

follows. Integrating the right component yields the general solution

y=12expC1x+16x3exp2C1x13x3dx+12C21/2

of Eq. (18). It does not have a Lie symmetry. □

The next example deals with a problem in hydrodynamics. The boundary layer at a circular cylinder immersed in the uniform flow of liquid is considered [14], see also Eq. 6.210 of [7].

Example 8 The equation

y3y+yy3y2y2+y2=0E20

has the only nonvanishing coefficient c2=3y21yy2+1. Substitution into (15) yields

(an+1abyn+bb2yna2y2n11y4yy2+1a2y2n+2abyn+1+b2y2=0.E21

It turns out that for n=3this condition specializes to

(a4a2+2aby3+b+4a28ab+2b2y4ab2yy2+1=0.

After some simplifications the resulting system for aand bis a2a2=0and b=a; Its solution a=b=12x+Cleads to the Bernoulli equation y12x+Cy312x+C=0with general solution

y=1C1xC2C1x+C2.

This is also the general solution of Eq. (20)

In general it is a priori not known whether there exists a Bernoulli component of any order. If a component for small values of ncannot be found it is desirable to determine bounds for its possible existence. The next example shows that this is possible in special cases.

Example 9 Consider the equation

y+xy1y2+y+xy=0E22

with K=2and non-vanishing coefficients c2=xy1, c1=1and c0=xy. Substitution into (15) yields

an+1abyn+bb2yna2y2n1xy+ayn+byxy1ayn+by2.

Expanding the last term into unique partial fractions by using the general formula

ynyk=ν=0n1knν1yν+knykE23

leads to

an+1abyn+bb2yna2y2n1xy+ayn+bya2xν=02n1yν2abxν=0nyνa+b2xy1=0.

The coefficients of the various terms yield a system for the unknowns a, band n. There is always the subsystem a+a2+a+x=0, a+b=0independent of n, it originates from the coefficient of yand the term independent of y. Furthermore, the leading term of the first sum in the above equation requires a=0for for any n2. These equations combined are inconsistent, i.e. the above Eq. (22) does not allow a Bernoulli component for any nonnegative natural number n. A similar reasoning exists for negative values of n.

At the moment an algorithm for determining bounds for nis not known, it is not even clear whether the existence of bounds is decidable in general.

3. Equations with leading term yy

Another important class of differential equations are those with leading term yy, they are considered in this section. Their general form is

yy+cxyy+k=0Kckxyyk=0withcxy,ckxyQxy,KN.E24

Components of Clairaut or d’Alembert type zyxfygymay lead to partial or even general solutions in closed form, mostly in a parameter representation. The main result of this section is given in the following proposition.

Proposition 4 Let a second-order differential Eq. (24) be given. A first-order component zyxfygyexists if fyand gysatisfy

pfpcxxfp+gp+p+xfp+gpk=0Kckxxfp+gppk=0E25

where pyhas been defined. Representing the left hand side of (25) as a partial fraction w.r.t. xand equating the coefficients of the various terms to zero, a system of first-order quasilinear ode’s for fpand gpis obtained; its degree in fpand gpis not higher than the degree in yof the coefficients cxyand ckxy.

Proof. Reduction of (24) w.r.t. zyxfygyleads to Eq. (25). Their properties follow directly from the assumptions about the coefficients cxyand ckxyin (24), and representing the left hand side of (25) as a partial fraction in x.

The determining system for the two functions fpand gpmay be obtained explicitly from (25) if the coefficients cxyand ckxyare known. Without restrictions on the coefficients ckxythe derivatives fpand gpmay occur linearly in any equation obtained after separation w.r.t. x, and an algebraic system in p, fp, gp, fpand gpfollows. It turns out that an algebraic Gröbner basis algorithm including factorization is a suitable tool for solving them in many cases. If a solution has been obtained the corresponding component may be applied for generating the decomposition of the given equation explicitly. The following example uses this proceeding.

Example 10 Consider the equation

yy+12yy+x12xy212xyy+12x=0.E26

Here J=1and K=2, its nonvanishing coefficients are cxy=12y, c2=x12x, c1=y2xand c0=12x. A linear or Bernoulli component does not exist. Proposition 4 leads to the system

ggp+gp2g=0,ffpfp2+f2fp=0,fgp+fp2f+fgp+fg+2fpgp2gp2p2=0.

Transforming the left-hand sides into algebraic Gröbner bases in the term order fp>gp>fp>gp>p, the following two systems and their solutions are obtained.

fp=0,gp+p21=0,f=p,g=1pp,fp+f=0,gp+p21=0,f=C1p,g=1pp.

They lead to the decompositions

zz+xy2+y2+1yz+xy2y21xyzzyxy+y1y=0,zz+y2+Cx+1yzxy2+y2+1xyzzyCyx+y1y=0,

respectively. The former decomposition generates a Clairaut component. It yields the solution y=Cx+1CCof (26), Cis an undetermined constant. Its parameter solution x=1p2+1, y=2pdoes not solve it, it annihilates a lower-order factor of the expression in the left-hand bracket of this decomposition and has to be discarded.

Integration of the d’Alembert component zy2+yyCx1leads to the general solution of (26) in a parameter representation

x=1C1C1+p2C1+p2C1plogC1+p2+pC1+C1C2+1p,y=1C1+p2pC1+p2C1logC1+p2+pC1+C1C2+1.

Eq. (26) does not have a Lie symmetry. □

This example shows that solutions of a component must be tested to see if they meet the second order equation, otherwise they have to be discarded; this phenomenon seems to be quite common.

4. Conclusions

The structure of the determining systems for linear or Bernoulli components of a nonlinear Eq. (2) given in Propositions 2 or 3, respectively, show clearly its relation to the corresponding system for the decomposition of a linear equation. For a generic equation of the second order this appears to be the best possible result. The same applies to the verious solution steps given on page 6. The corresponding result for determining Clairaut and d’Alembert components given in Proposition 4 is less specific. However, it should be possible, to obtain more detailed results if special classes of second-order equations are considered. In general, this area is only at an early stage and a better understanding of the underlying mechanisms generating the solutions and also its limitations would be highly desirable.

There are numerous possible generalizations fairly obvious. On the one hand, this concerns the equations to be solved. More general function fields for its coefficients like e.g. algebraic or elementary functions may be allowed. Equations of order three or four would be interesting in many applications. The greatest challenge however is certainly to develop similar procedures for partial differential equations as it has been indicated in Section 5 of [6].

On the other hand, the component type offers space for extensions too. In principle all equations of first order, as described for example in Kamke’s book [7], Part A, Section 4, are possible components. Components that guarantee at least a partial solution are of course particularly useful, the most important of them have been discussed in this article.

In order to apply decompositions to concrete problems the implementation of the procedures described in this article are available on the websitewww.alltypes.de[15].

Beyond that there are a number of general problems related to decompositions. For instance the question how rare are equations that allow a particular decomposition, Example 6 provides a partial answer. If two or more one-parameter families of solution curves are known as in Example 4, does it faciliate generating the general solution? The exact relation between Lie’s symmetry analysis and solution by decompositions is another subject of interest.

Acknowledgments

Various helpful comments of a referee are gratefully acknowledged.

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Fritz Schwarz (January 4th 2021). Solving Second-Order Differential Equations by Decomposition [Online First], IntechOpen, DOI: 10.5772/intechopen.94993. Available from:

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