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1 Algebraic and Analytic MethodsTopics of Discussion

§1.13 Differential Equations

  1. §1.13(i) Existence of Solutions
  2. §1.13(ii) Equations with a Parameter
  3. §1.13(iii) Inhomogeneous Equations
  4. §1.13(iv) Change of Variables
  5. §1.13(v) Products of Solutions
  6. §1.13(vi) Singularities
  7. §1.13(vii) Closed-Form Solutions
  8. §1.13(viii) Eigenvalues and Eigenfunctions: Sturm-Liouville and Liouville forms

§1.13(i) Existence of Solutions

A domain in the complex plane is simply-connected if it has no “holes”; more precisely, if its complement in the extended plane {} is connected.

The equation

1.13.1 d2wdz2+f(z)dwdz+g(z)w=0,

where zD, a simply-connected domain, and f(z), g(z) are analytic in D, has an infinite number of analytic solutions in D. A solution becomes unique, for example, when w and dw/dz are prescribed at a point in D.

Fundamental Pair

Two solutions w1(z) and w2(z) are called a fundamental pair if any other solution w(z) is expressible as

1.13.2 w(z)=Aw1(z)+Bw2(z),

where A and B are constants. A fundamental pair can be obtained, for example, by taking any z0D and requiring that

1.13.3 w1(z0) =1,
w1(z0) =0,
w2(z0) =0,
w2(z0) =1.


The Wronskian of w1(z) and w2(z) is defined by

1.13.4 𝒲{w1(z),w2(z)}=det[w1(z)w2(z)w1(z)w2(z)]=w1(z)w2(z)w2(z)w1(z).

(More generally 𝒲{w1(z),,wn(z)}=det[wk(j1)(z)], where 1j,kn.) Then the following relation is known as Abel’s identity

1.13.5 𝒲{w1(z),w2(z)}=cef(z)dz,

where c is independent of z and f(z) is defined in (1.13.1). (More generally in (1.13.5) for nth-order differential equations, f(z) is the coefficient multiplying the (n1)th-order derivative of the solution divided by the coefficient multiplying the nth-order derivative of the solution, see Ince (1926, §5.2).) If f(z)=0, then the Wronskian is constant.

The following three statements are equivalent: w1(z) and w2(z) comprise a fundamental pair in D; 𝒲{w1(z),w2(z)} does not vanish in D; w1(z) and w2(z) are linearly independent, that is, the only constants A and B such that

1.13.6 Aw1(z)+Bw2(z)=0,

are A=B=0.

§1.13(ii) Equations with a Parameter

Assume that in the equation

1.13.7 d2wdz2+f(u,z)dwdz+g(u,z)w=0,

u and z belong to domains U and D respectively, the coefficients f(u,z) and g(u,z) are continuous functions of both variables, and for each fixed u (fixed z) the two functions are analytic in z (in u). Suppose also that at (a fixed) z0D, w and w/z are analytic functions of u. Then at each zD, w, w/z and 2w/z2 are analytic functions of u.

§1.13(iii) Inhomogeneous Equations

The inhomogeneous (or nonhomogeneous) equation

1.13.8 d2wdz2+f(z)dwdz+g(z)w=r(z)

with f(z), g(z), and r(z) analytic in D has infinitely many analytic solutions in D. If w0(z) is any one solution, and w1(z), w2(z) are a fundamental pair of solutions of the corresponding homogeneous equation (1.13.1), then every solution of (1.13.8) can be expressed as

1.13.9 w(z)=w0(z)+Aw1(z)+Bw2(z),

where A and B are constants.

Variation of Parameters

With the notation of (1.13.8) and (1.13.9)

1.13.10 w0(z)=w2(z)w1(z)r(z)𝒲{w1(z),w2(z)}dzw1(z)w2(z)r(z)𝒲{w1(z),w2(z)}dz.

§1.13(iv) Change of Variables

Transformation of the Point at Infinity

The substitution ξ=1/z in (1.13.1) gives

1.13.11 d2Wdξ2+F(ξ)dWdξ+G(ξ)W=0,


1.13.12 W(ξ) =w(1ξ),
F(ξ) =2ξ1ξ2f(1ξ),
G(ξ) =1ξ4g(1ξ).

Elimination of First Derivative by Change of Dependent Variable

The substitution

1.13.13 w(z)=W(z)exp(12f(z)dz)

in (1.13.1) gives

1.13.14 d2Wdz2H(z)W=0,


1.13.15 H(z)=14f2(z)+12f(z)g(z).

Elimination of First Derivative by Change of Independent Variable

In (1.13.1) substitute

1.13.16 η=exp(f(z)dz)dz.


1.13.17 d2wdη2+g(z)exp(2f(z)dz)w=0.

Liouville Transformation

Let W(z) satisfy (1.13.14), ζ(z) be any thrice-differentiable function of z, and

1.13.18 U(z)=(ζ(z))1/2W(z).


1.13.19 d2Udζ2=(z˙2H(z)12{z,ζ})U.

Here dots denote differentiations with respect to ζ, and {z,ζ} is the Schwarzian derivative:

1.13.20 {z,ζ}=2z˙1/2d2dζ2(z˙1/2)=z˙˙˙z˙32(z¨z˙)2.

Cayley’s Identity

For arbitrary ξ and ζ,

1.13.21 {z,ζ} =(dξ/dζ)2{z,ξ}+{ξ,ζ}.
1.13.22 {z,ζ} =(dz/dζ)2{ζ,z}.

§1.13(v) Products of Solutions

The product of any two solutions of (1.13.1) satisfies

1.13.23 d3wdz3+3fd2wdz2+(2f2+f+4g)dwdz+(4fg+2g)w=0.

If U(z) and V(z) are respectively solutions of

1.13.24 d2Udz2+IU =0,
d2Vdz2+JV =0,

then W=UV is a solution of

1.13.25 ddz(W′′′+2(I+J)W+(I+J)WIJ)=(IJ)W.

For extensions of these results to linear homogeneous differential equations of arbitrary order see Spigler (1984).

§1.13(vi) Singularities

For classification of singularities of (1.13.1) and expansions of solutions in the neighborhoods of singularities, see §2.7.

§1.13(vii) Closed-Form Solutions

For an extensive collection of solutions of differential equations of the first, second, and higher orders see Kamke (1977).

§1.13(viii) Eigenvalues and Eigenfunctions: Sturm-Liouville and Liouville forms

A standard form for second order ordinary differential equations with x, and with a real parameter λ, and real valued functions p(x),q(x), and ρ(x), with p(x) and ρ(x) positive, is

1.13.26 (p(x)u(x))+(λρ(x)q(x))u(x)=0.

This is the Sturm-Liouville form of a second order differential equation, where denotes ddx. Assuming that u(x) satisfies un-mixed boundary conditions of the form

1.13.27 αu(a)+αu(a) =0,
α, α not both zero,
βu(b)+βu(b) =0,
β, β not both zero,

or periodic boundary conditions

1.13.28 u(a) =u(b),
u(a) =u(b),

on a finite interval [a,b], this is then a regular Sturm-Liouville system.

Eigenvalues and Eigenfunctions

A regular Sturm-Liouville system will only have solutions for certain (real) values of λ, these are eigenvalues. The functions u(x) which correspond to these being eigenfunctions. See for example Birkhoff and Rota (1989, Ch. 10) and the overview of Amrein et al. (2005).

Transformation to Liouville normal Form

Equation (1.13.26) with x[a,b] may be transformed to the Liouville normal form

1.13.29 w¨(t)+(λq^(t))w(t)=0,

where w¨ now denotes d2wdt2, via the transformation

1.13.30 w(t) =u(x)(p(x)ρ(x))1/4,
t =axρ(s)/p(s)ds,

and where

1.13.31 q^(t)=q/ρ+(pρ)1/4d2dt2(pρ)1/4.

As the interval [a,b] is mapped, one-to-one, onto [0,c] by the above definition of t, the integrand being positive, the inverse of this same transformation allows q^(t) to be calculated from p,q,ρ in (1.13.31), p,ρC2(a,b) and qC(a,b).

For a regular Sturm-Liouville system, equations (1.13.26) and (1.13.29) have: (i) identical eigenvalues, λ; (ii) the corresponding (real) eigenfunctions, u(x) and w(t), have the same number of zeros, also called nodes, for t(0,c) as for x(a,b); (iii) the eigenfunctions also satisfy the same type of boundary conditions, un-mixed or periodic, for both forms at the corresponding boundary points. See Birkhoff and Rota (1989, §§10.9, 10.10), Everitt (1982, §4.3), Olver (1997b, Ch. 6).