# §1.13 Differential Equations

## §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 $\mathbb{C}\cup\{\infty\}$ is connected.

The equation

 1.13.1 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+f(z)\frac{\mathrm{d}w}{\mathrm{d}z% }+g(z)w=0,$

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 $\ifrac{\mathrm{d}w}{\mathrm{d}z}$ are prescribed at a point in $D$.

### Fundamental Pair

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 $w(z)=Aw_{1}(z)+Bw_{2}(z),$

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 $\displaystyle w_{1}(z_{0})$ $\displaystyle=1,$ $\displaystyle w_{1}^{\prime}(z_{0})$ $\displaystyle=0,$ $\displaystyle w_{2}(z_{0})$ $\displaystyle=0,$ $\displaystyle w_{2}^{\prime}(z_{0})$ $\displaystyle=1.$ ⓘ Symbols: $z$: variable, $w_{1}(z)$: solution and $w_{2}(z)$: solution Permalink: http://dlmf.nist.gov/1.13.E3 Encodings: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png See also: Annotations for §1.13(i), §1.13(i), §1.13 and Ch.1

### Wronskian

The Wronskian of $w_{1}(z)$ and $w_{2}(z)$ is defined by

 1.13.4 $\mathscr{W}\left\{w_{1}(z),w_{2}(z)\right\}=\det\begin{bmatrix}w_{1}(z)&w_{2}(% z)\\ w_{1}^{\prime}(z)&w_{2}^{\prime}(z)\end{bmatrix}=w_{1}(z)w_{2}^{\prime}(z)-w_{% 2}(z)w_{1}^{\prime}(z).$ ⓘ Symbols: $\mathscr{W}$: Wronskian, $\det$: determinant, $z$: variable, $w_{1}(z)$: solution and $w_{2}(z)$: solution Referenced by: §1.13(i), Erratum (V1.0.25) for Section 1.13 Permalink: http://dlmf.nist.gov/1.13.E4 Encodings: TeX, pMML, png Addition (effective with 1.0.25): The determinant form of the two-argument Wronskian was added as an equality to this equation. See also: Annotations for §1.13(i), §1.13(i), §1.13 and Ch.1

(More generally $\mathscr{W}\left\{w_{1}(z),\ldots,w_{n}(z)\right\}=\det\left[w_{k}^{(j-1)}(z)\right]$, where $1\leq j,k\leq n$.) Then the following relation is known as Abel’s identity

 1.13.5 $\mathscr{W}\left\{w_{1}(z),w_{2}(z)\right\}=c{\mathrm{e}}^{-\int f(z)\,\mathrm% {d}z},$

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 $D$; $\mathscr{W}\left\{w_{1}(z),w_{2}(z)\right\}$ 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 $Aw_{1}(z)+Bw_{2}(z)=0,$ $\forall z\in D$,

are $A=B=0$.

## §1.13(ii) Equations with a Parameter

Assume that in the equation

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

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

## §1.13(iii) Inhomogeneous Equations

The inhomogeneous (or nonhomogeneous) equation

 1.13.8 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+f(z)\frac{\mathrm{d}w}{\mathrm{d}z% }+g(z)w=r(z)$

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 $w(z)=w_{0}(z)+Aw_{1}(z)+Bw_{2}(z),$

where $A$ and $B$ are constants.

### Variation of Parameters

With the notation of (1.13.8) and (1.13.9)

 1.13.10 $w_{0}(z)=w_{2}(z)\int\frac{w_{1}(z)r(z)}{\mathscr{W}\left\{w_{1}(z),w_{2}(z)% \right\}}\,\mathrm{d}z-w_{1}(z)\int\frac{w_{2}(z)r(z)}{\mathscr{W}\left\{w_{1}% (z),w_{2}(z)\right\}}\,\mathrm{d}z.$

## §1.13(iv) Change of Variables

### Transformation of the Point at Infinity

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

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

where

 1.13.12 $\displaystyle W(\xi)$ $\displaystyle=w\left(\frac{1}{\xi}\right),$ $\displaystyle F(\xi)$ $\displaystyle=\frac{2}{\xi}-\frac{1}{\xi^{2}}f\left(\frac{1}{\xi}\right),$ $\displaystyle G(\xi)$ $\displaystyle=\frac{1}{\xi^{4}}g\left(\frac{1}{\xi}\right).$

### Elimination of First Derivative by Change of Dependent Variable

The substitution

 1.13.13 $w(z)=W(z)\exp\left(-\tfrac{1}{2}\int f(z)\,\mathrm{d}z\right)$

in (1.13.1) gives

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

where

 1.13.15 $H(z)=\tfrac{1}{4}f^{2}(z)+\tfrac{1}{2}f^{\prime}(z)-g(z).$ ⓘ Defines: $H(z)$ (locally) Symbols: $z$: variable, $f(z)$: analytic coefficient and $g(z)$: analytic coefficient Permalink: http://dlmf.nist.gov/1.13.E15 Encodings: TeX, pMML, png See also: Annotations for §1.13(iv), §1.13(iv), §1.13 and Ch.1

### Elimination of First Derivative by Change of Independent Variable

In (1.13.1) substitute

 1.13.16 $\eta=\int\exp\left(-\int f(z)\,\mathrm{d}z\right)\,\mathrm{d}z.$ ⓘ Defines: $\eta$: change of variable (locally) Symbols: $\,\mathrm{d}\NVar{x}$: differential of $x$, $\exp\NVar{z}$: exponential function, $\int$: integral, $z$: variable and $f(z)$: analytic coefficient Permalink: http://dlmf.nist.gov/1.13.E16 Encodings: TeX, pMML, png See also: Annotations for §1.13(iv), §1.13(iv), §1.13 and Ch.1

Then

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

### Liouville Transformation

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

 1.13.18 $U(z)=(\zeta^{\prime}(z))^{1/2}W(z).$ ⓘ Symbols: $z$: variable, $W(\xi)$: solution, $\zeta(z)$: thrice-differentiable function and $U(z)$: solution Permalink: http://dlmf.nist.gov/1.13.E18 Encodings: TeX, pMML, png See also: Annotations for §1.13(iv), §1.13(iv), §1.13 and Ch.1

Then

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

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

 1.13.20 $\left\{z,\zeta\right\}=-2\dot{z}^{\ifrac{1}{2}}\frac{{\mathrm{d}}^{2}}{{% \mathrm{d}\zeta}^{2}}(\dot{z}^{-\ifrac{1}{2}})=\frac{\dddot{z}}{\dot{z}}-\frac% {3}{2}\left(\frac{\ddot{z}}{\dot{z}}\right)^{2}.$

### Cayley’s Identity

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

 1.13.21 $\displaystyle\left\{z,\zeta\right\}$ $\displaystyle=(\ifrac{\mathrm{d}\xi}{\mathrm{d}\zeta})^{2}\left\{z,\xi\right\}% +\left\{\xi,\zeta\right\}.$ 1.13.22 $\displaystyle\left\{z,\zeta\right\}$ $\displaystyle=-(\ifrac{\mathrm{d}z}{\mathrm{d}\zeta})^{2}\left\{\zeta,z\right\}.$

## §1.13(v) Products of Solutions

The product of any two solutions of (1.13.1) satisfies

 1.13.23 $\frac{{\mathrm{d}}^{3}w}{{\mathrm{d}z}^{3}}+3f\frac{{\mathrm{d}}^{2}w}{{% \mathrm{d}z}^{2}}+(2f^{2}+f^{\prime}+4g)\frac{\mathrm{d}w}{\mathrm{d}z}+(4fg+2% g^{\prime})w=0.$

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

 1.13.24 $\displaystyle\frac{{\mathrm{d}}^{2}U}{{\mathrm{d}z}^{2}}+IU$ $\displaystyle=0,$ $\displaystyle\frac{{\mathrm{d}}^{2}V}{{\mathrm{d}z}^{2}}+JV$ $\displaystyle=0,$

then $W=UV$ is a solution of

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

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

## §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 $x\in\mathbb{R}$, and with a real parameter $\lambda$, and real valued functions $p(x),q(x),$ and $\rho(x)$, with $p(x)$ and $\rho(x)$ positive, is

 1.13.26 $\left(p(x)u^{\prime}(x)\right)^{\prime}+\left(\lambda\rho(x)-q(x)\right)u(x)=0.$ ⓘ Symbols: $p(x)$: function, $q(x)$: function, $\rho(x)$: function, $u(x)$: solution, $\lambda$: real parameter and $x$: real variable Referenced by: §1.13(viii), §1.13(viii), Erratum (V1.2.0) §1.13 Permalink: http://dlmf.nist.gov/1.13.E26 Encodings: TeX, pMML, png See also: Annotations for §1.13(viii), §1.13 and Ch.1

This is the Sturm-Liouville form of a second order differential equation, where denotes $\frac{\mathrm{d}}{\mathrm{d}x}$. Assuming that $u(x)$ satisfies un-mixed boundary conditions of the form

 1.13.27 $\displaystyle\alpha u(a)+\alpha^{\prime}u^{\prime}(a)$ $\displaystyle=0,$ $\alpha$, $\alpha^{\prime}$ not both zero, $\displaystyle\beta u(b)+\beta^{\prime}u^{\prime}(b)$ $\displaystyle=0,$ $\beta$, $\beta^{\prime}$ not both zero,

or periodic boundary conditions

 1.13.28 $\displaystyle u(a)$ $\displaystyle=u(b),$ $\displaystyle u^{\prime}(a)$ $\displaystyle=u^{\prime}(b),$ ⓘ Symbols: $u(x)$: solution, $a$: real variable and $b$: real variable Permalink: http://dlmf.nist.gov/1.13.E28 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §1.13(viii), §1.13 and Ch.1

on a finite interval $[a,b]\subset\mathbb{R}$, 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 $\lambda$, 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 $x\in[a,b]$ may be transformed to the Liouville normal form

 1.13.29 $\ddot{w}(t)+\left(\lambda-\widehat{q}(t)\right)w(t)=0,$ $t\in[0,c]$

where $\ddot{w}$ now denotes $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}t}^{2}}$, via the transformation

 1.13.30 $\displaystyle w(t)$ $\displaystyle=u(x)\left(p(x)\rho(x)\right)^{1/4},$ $\displaystyle t$ $\displaystyle=\int_{a}^{x}\sqrt{\rho{(s)}/p(s)}\,\mathrm{d}s,$

and where

 1.13.31 $\widehat{q}(t)=q/\rho+\left(p\rho\right)^{-1/4}\frac{{\mathrm{d}}^{2}}{{% \mathrm{d}t}^{2}}\left(p\rho\right)^{1/4}.$

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 $\widehat{q}(t)$ to be calculated from $p,q,\rho$ in (1.13.31), $p,\rho\in C^{2}(a,b)$ and $q\in C(a,b)$.

For a regular Sturm-Liouville system, equations (1.13.26) and (1.13.29) have: (i) identical eigenvalues, $\lambda$; (ii) the corresponding (real) eigenfunctions, $u(x)$ and $w(t)$, have the same number of zeros, also called nodes, for $t\in(0,c)$ as for $x\in(a,b)$; (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).