- §1.13(i) Existence of Solutions
- §1.13(ii) Equations with a Parameter
- §1.13(iii) Inhomogeneous Equations
- §1.13(iv) Change of Variables
- §1.13(v) Products of Solutions
- §1.13(vi) Singularities
- §1.13(vii) Closed-Form Solutions

A domain in the complex plane is *simply-connected*
if it has no “holes”; more precisely, if its complement in the extended plane
$\u2102\cup \{\mathrm{\infty}\}$ is connected.

The equation

1.13.1 | $$\frac{{d}^{2}w}{{dz}^{2}}+f(z)\frac{dw}{dz}+g(z)w=0,$$ | ||

where $z\in D$, 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 $D$.

Two solutions ${w}_{1}(z)$ and ${w}_{2}(z)$ are called a *fundamental pair*
if any other solution $w(z)$ is expressible as

1.13.2 | $$w(z)=A{w}_{1}(z)+B{w}_{2}(z),$$ | ||

where $A$ and $B$ are constants. A fundamental pair can be obtained, for example, by taking any ${z}_{0}\in D$ and requiring that

1.13.3 | ${w}_{1}({z}_{0})$ | $=1,$ | ||

${w}_{1}^{\prime}({z}_{0})$ | $=0,$ | |||

${w}_{2}({z}_{0})$ | $=0,$ | |||

${w}_{2}^{\prime}({z}_{0})$ | $=1.$ | |||

The *Wronskian* of ${w}_{1}(z)$ and ${w}_{2}(z)$ is defined by

1.13.4 | $$\mathcal{W}\left\{{w}_{1}(z),{w}_{2}(z)\right\}=det\left[\begin{array}{cc}{w}_{1}(z)& {w}_{2}(z)\\ {w}_{1}^{\prime}(z)& {w}_{2}^{\prime}(z)\end{array}\right]={w}_{1}(z){w}_{2}^{\prime}(z)-{w}_{2}(z){w}_{1}^{\prime}(z).$$ | ||

(More generally $\mathcal{W}\left\{{w}_{1}(z),\mathrm{\dots},{w}_{n}(z)\right\}=det\left[{w}_{k}^{(j-1)}(z)\right]$,
where $1\le j,k\le n$.) Then the following relation is known as *Abel’s identity*

1.13.5 | $$\mathcal{W}\left\{{w}_{1}(z),{w}_{2}(z)\right\}=c{\mathrm{e}}^{-{\scriptscriptstyle \int f(z)dz}},$$ | ||

where $c$ is independent of $z$ and $f(z)$ is defined in (1.13.1). (More generally in (1.13.5) for $n$th-order differential equations, $f(z)$ is the coefficient multiplying the $(n-1)$th-order derivative of the solution divided by the coefficient multiplying the $n$th-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: ${w}_{1}(z)$ and ${w}_{2}(z)$ comprise a
fundamental pair in $D$; $\mathcal{W}\left\{{w}_{1}(z),{w}_{2}(z)\right\}$ does not vanish in $D$;
${w}_{1}(z)$ and ${w}_{2}(z)$ are *linearly independent*,
that is, the only constants $A$ and $B$ such that

1.13.6 | $$A{w}_{1}(z)+B{w}_{2}(z)=0,$$ | ||

$\forall z\in D$, | |||

are $A=B=0$.

Assume that in the equation

1.13.7 | $$\frac{{d}^{2}w}{{dz}^{2}}+f(u,z)\frac{dw}{dz}+g(u,z)w=0,$$ | ||

$u$ 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) ${z}_{0}\in D$, $w$ and $\partial w/\partial z$ are analytic functions of $u$. Then at each $z\in D$, $w$, $\partial w/\partial z$ and ${\partial}^{2}w/{\partial z}^{2}$ are analytic functions of $u$.

The *inhomogeneous*
(or *nonhomogeneous*)
equation

1.13.8 | $$\frac{{d}^{2}w}{{dz}^{2}}+f(z)\frac{dw}{dz}+g(z)w=r(z)$$ | ||

with $f(z)$, $g(z)$, and $r(z)$ analytic in $D$ has infinitely many analytic solutions in $D$. If ${w}_{0}(z)$ is any one solution, and ${w}_{1}(z)$, ${w}_{2}(z)$ 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 | $$w(z)={w}_{0}(z)+A{w}_{1}(z)+B{w}_{2}(z),$$ | ||

where $A$ and $B$ are constants.

The substitution $\xi =1/z$ in (1.13.1) gives

1.13.11 | $$\frac{{d}^{2}W}{{d\xi}^{2}}+F(\xi )\frac{dW}{d\xi}+G(\xi )W=0,$$ | ||

where

1.13.12 | $W(\xi )$ | $=w\left({\displaystyle \frac{1}{\xi}}\right),$ | ||

$F(\xi )$ | $={\displaystyle \frac{2}{\xi}}-{\displaystyle \frac{1}{{\xi}^{2}}}f\left({\displaystyle \frac{1}{\xi}}\right),$ | |||

$G(\xi )$ | $={\displaystyle \frac{1}{{\xi}^{4}}}g\left({\displaystyle \frac{1}{\xi}}\right).$ | |||

The substitution

1.13.13 | $$w(z)=W(z)\mathrm{exp}\left(-\frac{1}{2}\int f(z)dz\right)$$ | ||

in (1.13.1) gives

1.13.14 | $$\frac{{d}^{2}W}{{dz}^{2}}-H(z)W=0,$$ | ||

where

1.13.15 | $$H(z)=\frac{1}{4}{f}^{2}(z)+\frac{1}{2}{f}^{\prime}(z)-g(z).$$ | ||

In (1.13.1) substitute

1.13.16 | $$\eta =\int \mathrm{exp}\left(-\int f(z)dz\right)dz.$$ | ||

Then

1.13.17 | $$\frac{{d}^{2}w}{{d\eta}^{2}}+g(z)\mathrm{exp}\left(2\int f(z)dz\right)w=0.$$ | ||

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

1.13.18 | $$U(z)={({\zeta}^{\prime}(z))}^{1/2}W(z).$$ | ||

Then

1.13.19 | $$\frac{{d}^{2}U}{{d\zeta}^{2}}=\left({\dot{z}}^{2}H(z)-\frac{1}{2}\{z,\zeta \}\right)U.$$ | ||

Here dots denote differentiations with respect to $\zeta $, and
$\{z,\zeta \}$ is the *Schwarzian derivative*:

1.13.20 | $$\{z,\zeta \}=-2{\dot{z}}^{1/2}\frac{{d}^{2}}{{d\zeta}^{2}}({\dot{z}}^{-1/2})=\frac{\stackrel{\dot{}\dot{}\dot{}}{z}}{\dot{z}}-\frac{3}{2}{\left(\frac{\ddot{z}}{\dot{z}}\right)}^{2}.$$ | ||

For arbitrary $\xi $ and $\zeta $,

1.13.21 | $\{z,\zeta \}$ | $={(d\xi /d\zeta )}^{2}\{z,\xi \}+\{\xi ,\zeta \}.$ | ||

1.13.22 | $\{z,\zeta \}$ | $=-{(dz/d\zeta )}^{2}\{\zeta ,z\}.$ | ||

The product of any two solutions of (1.13.1) satisfies

1.13.23 | $$\frac{{d}^{3}w}{{dz}^{3}}+3f\frac{{d}^{2}w}{{dz}^{2}}+(2{f}^{2}+{f}^{\prime}+4g)\frac{dw}{dz}+(4fg+2{g}^{\prime})w=0.$$ | ||

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

1.13.24 | $\frac{{d}^{2}U}{{dz}^{2}}}+IU$ | $=0,$ | ||

$\frac{{d}^{2}V}{{dz}^{2}}}+JV$ | $=0,$ | |||

then $W=UV$ is a solution of

1.13.25 | $$\frac{d}{dz}\left(\frac{{W}^{\prime \prime \prime}+2(I+J){W}^{\prime}+({I}^{\prime}+{J}^{\prime})W}{I-J}\right)=-(I-J)W.$$ | ||

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

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