§1.13 Differential Equations

Referenced by:: §3.7(i)
Permalink:: http://dlmf.nist.gov/1.13

§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

§1.13(i) Existence of Solutions

Notes:: See Olver (1997b, pp. 141–142, 145–146).
Keywords:: differential equations, domain, simply-connected domain
Referenced by:: §3.3(i), §31.9(i)
Permalink:: http://dlmf.nist.gov/1.13.i

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

The equation

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

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $z$ : variable and $w(z)$ : solution
Referenced by:: ¶ ‣ §1.13(iv), ¶ ‣ §1.13(iv), ¶ ‣ §1.13(iv), §1.13(iii), §1.13(v), §1.13(vi)
Permalink:: http://dlmf.nist.gov/1.13.E1
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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 $\ifrac{dw}{dz}$ are prescribed at a point in $D$ .

¶ Fundamental Pair

Keywords:: differential equations

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

Symbols:: $z$ : variable, $w(z)$ : solution, $A$ : constant and $B$ : constant
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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

$w_{1}(z_{0})=1,$

$w_{1}^{{\prime}}(z_{0})=0,$

$w_{2}(z_{0})=0,$

$w_{2}^{{\prime}}(z_{0})=1.$

Symbols:: $z$ : variable and $w(z)$ : solution
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¶ Wronskian

Keywords:: Wronskian, differential equations

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

1.13.4 $\mathop{\mathscr{W}\/}\nolimits\left\{ w_{1}(z),w_{2}(z)\right\}=w_{1}(z)w_{2}^{{\prime}}(z)-w_{2}(z)w_{1}^{{\prime}}(z).$

Defines:: $\mathop{\mathscr{W}\/}\nolimits$ : Wronskian
Symbols:: $z$ : variable and $w(z)$ : solution
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Then

1.13.5 $\mathop{\mathscr{W}\/}\nolimits\left\{ w_{1}(z),w_{2}(z)\right\}=ce^{{-\int f(z)dz}},$

Symbols:: $\mathop{\mathscr{W}\/}\nolimits$ : Wronskian, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $z$ : variable and $w(z)$ : solution
Referenced by:: §10.24, §10.28, §10.45, §10.5, §10.50
Permalink:: http://dlmf.nist.gov/1.13.E5
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where $c$ is independent of $z$ . 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 $D$ ; $\mathop{\mathscr{W}\/}\nolimits\left\{ w_{1}(z),w_{2}(z)\right\}$ 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 $Aw_{1}(z)+Bw_{2}(z)=0,$ $\forall z\in D$ ,

Symbols:: $\in$ : element of, $\forall$ : for every, $z$ : variable, $D$ : simply-connected domain, $w(z)$ : solution, $A$ : constant and $B$ : constant
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are $A=B=0$ .

§1.13(ii) Equations with a Parameter

Notes:: See Olver (1997b, pp. 146–147).
Keywords:: differential equations
Referenced by:: §10.15
Permalink:: http://dlmf.nist.gov/1.13.ii

Assume that in the equation

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

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ and $z$ : variable
Permalink:: http://dlmf.nist.gov/1.13.E7
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$u$ 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

Notes:: For (1.13.10) see Simmons (1972, pp. 90–92).
Keywords:: differential equations
Referenced by:: §9.12(i)
Permalink:: http://dlmf.nist.gov/1.13.iii

The inhomogeneous (or nonhomogeneous) equation

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

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $z$ : variable and $w(z)$ : solution
Referenced by:: ¶ ‣ §1.13(iii), §1.13(iii)
Permalink:: http://dlmf.nist.gov/1.13.E8
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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 $w(z)=w_{0}(z)+Aw_{1}(z)+Bw_{2}(z),$

Symbols:: $z$ : variable, $w(z)$ : solution, $B$ : constant and $A$ : constant
Referenced by:: ¶ ‣ §1.13(iii)
Permalink:: http://dlmf.nist.gov/1.13.E9
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where $A$ and $B$ are constants.

¶ Variation of Parameters

Keywords:: differential equations, variation of parameters

With the notation of (1.13.8) and (1.13.9)

1.13.10 $w_{0}(z)=w_{2}(z)\int\frac{w_{1}(z)r(z)}{\mathop{\mathscr{W}\/}\nolimits\left\{ w_{1}(z),w_{2}(z)\right\}}dz-w_{1}(z)\int\frac{w_{2}(z)r(z)}{\mathop{\mathscr{W}\/}\nolimits\left\{ w_{1}(z),w_{2}(z)\right\}}dz.$

Symbols:: $\mathop{\mathscr{W}\/}\nolimits$ : Wronskian, $dx$ : differential of $x$ , $\int$ : integral, $z$ : variable and $w(z)$ : solution
Referenced by:: §1.13(iii)
Permalink:: http://dlmf.nist.gov/1.13.E10
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§1.13(iv) Change of Variables

Notes:: See Temme (1996b, pp. 84, 103), or Olver (1997b, pp. 190–191).
Referenced by:: §2.8(i)
Permalink:: http://dlmf.nist.gov/1.13.iv

¶ Transformation of the Point at Infinity

Keywords:: differential equations

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

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

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $\xi$ : change of variable, $W(\xi)$ , $F(\xi)$ and $G(\xi)$
Permalink:: http://dlmf.nist.gov/1.13.E11
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where

1.13.12

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

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

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

Symbols:: $w(z)$ : solution, $\xi$ : change of variable, $W(\xi)$ , $F(\xi)$ and $G(\xi)$
Permalink:: http://dlmf.nist.gov/1.13.E12
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¶ Elimination of First Derivative by Change of Dependent Variable

Keywords:: differential equations

The substitution

1.13.13 $w(z)=W(z)\mathop{\exp\/}\nolimits\!\left(-\tfrac{1}{2}\int f(z)dz\right)$

Symbols:: $dx$ : differential of $x$ , $\mathop{\exp\/}\nolimits z$ : exponential function, $\int$ : integral, $z$ : variable, $w(z)$ : solution and $W(z)$ : change of variable
Permalink:: http://dlmf.nist.gov/1.13.E13
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in (1.13.1) gives

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

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $z$ : variable, $W(z)$ : change of variable and $H(z)$
Referenced by:: ¶ ‣ §1.13(iv)
Permalink:: http://dlmf.nist.gov/1.13.E14
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where

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

Symbols:: $z$ : variable and $H(z)$
Permalink:: http://dlmf.nist.gov/1.13.E15
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¶ Elimination of First Derivative by Change of Independent Variable

Keywords:: differential equations

In (1.13.1) substitute

1.13.16 $\eta=\int\mathop{\exp\/}\nolimits\!\left(-\int f(z)dz\right)dz.$

Symbols:: $dx$ : differential of $x$ , $\mathop{\exp\/}\nolimits z$ : exponential function, $\int$ : integral, $z$ : variable and $\eta$ : change of variable
Permalink:: http://dlmf.nist.gov/1.13.E16
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Then

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

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $dx$ : differential of $x$ , $\mathop{\exp\/}\nolimits z$ : exponential function, $\int$ : integral, $z$ : variable, $w(z)$ : solution and $\eta$ : change of variable
Permalink:: http://dlmf.nist.gov/1.13.E17
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¶ Liouville Transformation

Keywords:: Liouville transformation for differential equations, Schwarzian derivative, differential equations

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, $W(z)$ : change of variable and $\zeta(z)$ : thrice-differentiable function
Permalink:: http://dlmf.nist.gov/1.13.E18
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Then

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

Symbols:: $\left\{ z,\zeta\right\}$ : Schwarzian derivative, $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $z$ : variable, $H(z)$ and $\zeta(z)$ : thrice-differentiable function
Permalink:: http://dlmf.nist.gov/1.13.E19
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Here dots denote differentiations with respect to $\zeta$ , and $\left\{ z,\zeta\right\}$ is the Schwarzian derivative:

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

Defines:: $\left\{ z,\zeta\right\}$ : Schwarzian derivative
Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $z$ : variable and $\zeta(z)$ : thrice-differentiable function
Permalink:: http://dlmf.nist.gov/1.13.E20
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¶ Cayley’s Identity

Keywords:: Cayley’s identity for Schwarzian derivatives

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

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

Symbols:: $\left\{ z,\zeta\right\}$ : Schwarzian derivative, $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $z$ : variable, $\xi$ : change of variable and $\zeta(z)$ : thrice-differentiable function
Permalink:: http://dlmf.nist.gov/1.13.E21
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1.13.22 $\left\{ z,\zeta\right\}=-(\ifrac{dz}{d\zeta})^{2}\left\{\zeta,z\right\}.$

Symbols:: $\left\{ z,\zeta\right\}$ : Schwarzian derivative, $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $z$ : variable and $\zeta(z)$ : thrice-differentiable function
Permalink:: http://dlmf.nist.gov/1.13.E22
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§1.13(v) Products of Solutions

Notes:: See Watson (1944, pp. 145–146).
Keywords:: differential equations
Referenced by:: §10.13, §9.11(i), Version 1.0.1 (June 27, 2011)
Permalink:: http://dlmf.nist.gov/1.13.v
Addition (effective with 1.0.1):: A sentence adding a reference was inserted at the end of this subsection.
Reported 2011-06-21