1 Algebraic and Analytic Methods Topics of Discussion 1.12 Continued Fractions 1.14 Integral Transforms

§1.13 Differential Equations

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Referenced by:: §3.7(i), Erratum (V1.0.25) for Section 1.13, Erratum (V1.2.0) §1.13
Permalink:: http://dlmf.nist.gov/1.13
See also:: Annotations for Ch.1

§1.13(i) Existence of Solutions

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Keywords:: differential equations, domain, existence, simply-connected, simply-connected domain, solutions
Notes:: See Olver (1997b, pp. 141–142, 145–146) and Ince (1926, §5.2).
Referenced by:: §3.3(i), §31.9(i), Erratum (V1.0.25) for Section 1.13
Permalink:: http://dlmf.nist.gov/1.13.i
See also:: Annotations for §1.13 and Ch.1

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		$\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+f(z)\frac{\mathrm{d}w}{\mathrm{d}z% }+g(z)w=0,$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : variable, $w(z)$ : solution, $f(z)$ : analytic coefficient and $g(z)$ : analytic coefficient Referenced by: §1.13(i), §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 Encodings: TeX, pMML, png See also: Annotations for §1.13(i), §1.13 and Ch.1

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

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Keywords:: differential equations, fundamental pair, solutions
See also:: Annotations for §1.13(i), §1.13 and Ch.1

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, $w_{1}(z)$ : solution, $w_{2}(z)$ : solution, $A$ : constant and $B$ : constant Permalink: http://dlmf.nist.gov/1.13.E2 Encodings: TeX, pMML, png See also: Annotations for §1.13(i), §1.13(i), §1.13 and Ch.1

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

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Defines:: $\mathscr{W}$ : Wronskian
Keywords:: Wronskian, differential equations, linearly independent, solutions
Referenced by:: Erratum (V1.0.25) for Section 1.13
Addition (effective with 1.0.25):: Immediately below Equation (1.13.4), a sentence was added giving the definition of the $n$ -argument Wronskian. It is explained just above (1.13.5) that this equation is often referred to as Abel’s identity. Immediately below Equation (1.13.5), a sentence was added explaining how it generalizes for $n$ th-order differential equations, with a reference to Ince (1926, §5.2).
See also:: Annotations for §1.13(i), §1.13 and Ch.1

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		$\mathscr{W}\left\{w_{1}(z),w_{2}(z)\right\}=c{\mathrm{e}}^{-\int f(z)\,\mathrm% {d}z},$
ⓘ Symbols: $\mathscr{W}$ : Wronskian, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $z$ : variable, $f(z)$ : analytic coefficient, $w_{1}(z)$ : solution and $w_{2}(z)$ : solution Referenced by: §1.13(i), §1.13(i), §10.24, §10.28, §10.45, §10.5, §10.50, Erratum (V1.0.25) for Section 1.13 Permalink: http://dlmf.nist.gov/1.13.E5 Encodings: TeX, pMML, png See also: Annotations for §1.13(i), §1.13(i), §1.13 and Ch.1

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		$Aw_{1}(z)+Bw_{2}(z)=0,$
$\forall z\in D$ ,
ⓘ Symbols: $\in$ : element of, $\forall$ : for every, $z$ : variable, $D$ : simply-connected domain, $w_{1}(z)$ : solution, $w_{2}(z)$ : solution, $A$ : constant and $B$ : constant Permalink: http://dlmf.nist.gov/1.13.E6 Encodings: TeX, pMML, png See also: Annotations for §1.13(i), §1.13(i), §1.13 and Ch.1

are $A=B=0$ .

§1.13(ii) Equations with a Parameter

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Keywords:: differential equations, with a parameter
Notes:: See Olver (1997b, pp. 146–147).
Referenced by:: §10.15
Permalink:: http://dlmf.nist.gov/1.13.ii
See also:: Annotations for §1.13 and Ch.1

Assume that in the equation

1.13.7		$\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+f(u,z)\frac{\mathrm{d}w}{\mathrm{d% }z}+g(u,z)w=0,$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : variable, $f(u,z)$ : analytic function of both variables, $g(u,z)$ : analytic function of both variables and $w(u,z)$ : solution Permalink: http://dlmf.nist.gov/1.13.E7 Encodings: TeX, pMML, png See also: Annotations for §1.13(ii), §1.13 and Ch.1

$u$ 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

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Keywords:: differential equations, homogeneous, inhomogeneous, nonhomogeneous
Notes:: For (1.13.10) see Simmons (1972, pp. 90–92).
Referenced by:: §9.12(i)
Permalink:: http://dlmf.nist.gov/1.13.iii
See also:: Annotations for §1.13 and Ch.1

The inhomogeneous (or nonhomogeneous) equation

1.13.8		$\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+f(z)\frac{\mathrm{d}w}{\mathrm{d}z% }+g(z)w=r(z)$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : variable, $w(z)$ : solution, $r(z)$ : analytic function, $f(z)$ : analytic coefficient and $g(z)$ : analytic coefficient Referenced by: §1.13(iii), §1.13(iii) Permalink: http://dlmf.nist.gov/1.13.E8 Encodings: TeX, pMML, png See also: Annotations for §1.13(iii), §1.13 and Ch.1

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, $w_{0}(z)$ : solution, $A$ : constant, $B$ : constant, $w_{1}(z)$ : solution and $w_{2}(z)$ : solution Referenced by: §1.13(iii) Permalink: http://dlmf.nist.gov/1.13.E9 Encodings: TeX, pMML, png See also: Annotations for §1.13(iii), §1.13 and Ch.1

where $A$ and $B$ are constants.

Variation of Parameters

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Keywords:: differential equations, inhomogeneous, inhomogeneous differential equations, solution by variation of parameters, variation of parameters
See also:: Annotations for §1.13(iii), §1.13 and Ch.1

With the notation of (1.13.8) and (1.13.9)

1.13.10		$w_{0}(z)=w_{2}(z)\int\frac{w_{1}(z)r(z)}{\mathscr{W}\left\{w_{1}(z),w_{2}(z)% \right\}}\,\mathrm{d}z-w_{1}(z)\int\frac{w_{2}(z)r(z)}{\mathscr{W}\left\{w_{1}% (z),w_{2}(z)\right\}}\,\mathrm{d}z.$
ⓘ Symbols: $\mathscr{W}$ : Wronskian, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $z$ : variable, $r(z)$ : analytic function, $w_{0}(z)$ : solution, $w_{1}(z)$ : solution and $w_{2}(z)$ : solution Referenced by: §1.13(iii) Permalink: http://dlmf.nist.gov/1.13.E10 Encodings: TeX, pMML, png See also: Annotations for §1.13(iii), §1.13(iii), §1.13 and Ch.1

§1.13(iv) Change of Variables

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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
See also:: Annotations for §1.13 and Ch.1

Transformation of the Point at Infinity

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Keywords:: change of variables, differential equations, point at infinity
See also:: Annotations for §1.13(iv), §1.13 and Ch.1

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

1.13.11		$\frac{{\mathrm{d}}^{2}W}{{\mathrm{d}\xi}^{2}}+F(\xi)\frac{\mathrm{d}W}{\mathrm% {d}\xi}+G(\xi)W=0,$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\xi$ : change of variable, $W(\xi)$ : solution, $F(\xi)$ : analytic coefficient and $G(\xi)$ : analytic coefficient Permalink: http://dlmf.nist.gov/1.13.E11 Encodings: TeX, pMML, png See also: Annotations for §1.13(iv), §1.13(iv), §1.13 and Ch.1

where

1.13.12	$\displaystyle W(\xi)$	$\displaystyle=w\left(\frac{1}{\xi}\right),$
	$\displaystyle F(\xi)$	$\displaystyle=\frac{2}{\xi}-\frac{1}{\xi^{2}}f\left(\frac{1}{\xi}\right),$
	$\displaystyle G(\xi)$	$\displaystyle=\frac{1}{\xi^{4}}g\left(\frac{1}{\xi}\right).$
ⓘ Symbols: $w(z)$ : solution, $f(z)$ : analytic coefficient, $\xi$ : change of variable, $W(\xi)$ : solution, $F(\xi)$ : analytic coefficient, $G(\xi)$ : analytic coefficient and $g(z)$ : analytic coefficient Permalink: http://dlmf.nist.gov/1.13.E12 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §1.13(iv), §1.13(iv), §1.13 and Ch.1

Elimination of First Derivative by Change of Dependent Variable

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Keywords:: change of variables, differential equations, elimination of first derivative
See also:: Annotations for §1.13(iv), §1.13 and Ch.1

The substitution

1.13.13		$w(z)=W(z)\exp\left(-\tfrac{1}{2}\int f(z)\,\mathrm{d}z\right)$
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\exp\NVar{z}$ : exponential function, $\int$ : integral, $z$ : variable, $w(z)$ : solution, $f(z)$ : analytic coefficient and $W(\xi)$ : solution Permalink: http://dlmf.nist.gov/1.13.E13 Encodings: TeX, pMML, png See also: Annotations for §1.13(iv), §1.13(iv), §1.13 and Ch.1

in (1.13.1) gives

1.13.14		$\frac{{\mathrm{d}}^{2}W}{{\mathrm{d}z}^{2}}-H(z)W=0,$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : variable, $W(\xi)$ : solution and $H(z)$ Referenced by: §1.13(iv) Permalink: http://dlmf.nist.gov/1.13.E14 Encodings: TeX, pMML, png See also: Annotations for §1.13(iv), §1.13(iv), §1.13 and Ch.1

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

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Keywords:: change of variables, differential equations, elimination of first derivative
See also:: Annotations for §1.13(iv), §1.13 and Ch.1

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		$\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}\eta}^{2}}+g(z)\exp\left(2\int f(z)\,% \mathrm{d}z\right)w=0.$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\exp\NVar{z}$ : exponential function, $\int$ : integral, $z$ : variable, $w(z)$ : solution, $f(z)$ : analytic coefficient, $\eta$ : change of variable and $g(z)$ : analytic coefficient Permalink: http://dlmf.nist.gov/1.13.E17 Encodings: TeX, pMML, png See also: Annotations for §1.13(iv), §1.13(iv), §1.13 and Ch.1

Liouville Transformation

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Keywords:: Liouville transformation, Liouville transformation for differential equations, Schwarzian derivative, change of variables, differential equations
See also:: Annotations for §1.13(iv), §1.13 and Ch.1

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		$\frac{{\mathrm{d}}^{2}U}{{\mathrm{d}\zeta}^{2}}=\left(\dot{z}^{2}H(z)-\tfrac{1% }{2}\left\{z,\zeta\right\}\right)U.$
ⓘ Symbols: $\left\{\NVar{z},\NVar{\zeta}\right\}$ : Schwarzian derivative, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : variable, $H(z)$ , $\zeta(z)$ : thrice-differentiable function, $U(z)$ : solution and $\dot{\NVar{z}}$ : derivative of $\NVar{z}$ with respect to $\zeta$ Permalink: http://dlmf.nist.gov/1.13.E19 Encodings: TeX, pMML, png See also: Annotations for §1.13(iv), §1.13(iv), §1.13 and Ch.1

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

1.13.20		$\left\{z,\zeta\right\}=-2\dot{z}^{\ifrac{1}{2}}\frac{{\mathrm{d}}^{2}}{{% \mathrm{d}\zeta}^{2}}(\dot{z}^{-\ifrac{1}{2}})=\frac{\dddot{z}}{\dot{z}}-\frac% {3}{2}\left(\frac{\ddot{z}}{\dot{z}}\right)^{2}.$
ⓘ Defines: $\left\{\NVar{z},\NVar{\zeta}\right\}$ : Schwarzian derivative Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : variable, $\zeta(z)$ : thrice-differentiable function, $\dot{\NVar{z}}$ : derivative of $\NVar{z}$ with respect to $\zeta$ , $\ddot{\NVar{z}}$ : 2nd derivative of $\NVar{z}$ with respect to $\zeta$ and $\dddot{\NVar{z}}$ : 3rd derivative of $\NVar{z}$ with respect to $\zeta$ Permalink: http://dlmf.nist.gov/1.13.E20 Encodings: TeX, pMML, png See also: Annotations for §1.13(iv), §1.13(iv), §1.13 and Ch.1

Cayley’s Identity

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Keywords:: Cayley’s identity for Schwarzian derivatives
See also:: Annotations for §1.13(iv), §1.13 and Ch.1

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

1.13.21	$\displaystyle\left\{z,\zeta\right\}$	$\displaystyle=(\ifrac{\mathrm{d}\xi}{\mathrm{d}\zeta})^{2}\left\{z,\xi\right\}% +\left\{\xi,\zeta\right\}.$
ⓘ Symbols: $\left\{\NVar{z},\NVar{\zeta}\right\}$ : Schwarzian derivative, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : 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 Encodings: TeX, pMML, png See also: Annotations for §1.13(iv), §1.13(iv), §1.13 and Ch.1
1.13.22	$\displaystyle\left\{z,\zeta\right\}$	$\displaystyle=-(\ifrac{\mathrm{d}z}{\mathrm{d}\zeta})^{2}\left\{\zeta,z\right\}.$
ⓘ Symbols: $\left\{\NVar{z},\NVar{\zeta}\right\}$ : Schwarzian derivative, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : variable and $\zeta(z)$ : thrice-differentiable function Permalink: http://dlmf.nist.gov/1.13.E22 Encodings: TeX, pMML, png See also: Annotations for §1.13(iv), §1.13(iv), §1.13 and Ch.1

§1.13(v) Products of Solutions

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Keywords:: differential equations, products, solutions
Notes:: See Watson (1944, pp. 145–146).
Referenced by:: §10.13, §9.11(i), Erratum (V1.0.1) for References
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.
See also:: Annotations for §1.13 and Ch.1

The product of any two solutions of (1.13.1) satisfies

1.13.23		$\frac{{\mathrm{d}}^{3}w}{{\mathrm{d}z}^{3}}+3f\frac{{\mathrm{d}}^{2}w}{{% \mathrm{d}z}^{2}}+(2f^{2}+f^{\prime}+4g)\frac{\mathrm{d}w}{\mathrm{d}z}+(4fg+2% g^{\prime})w=0.$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : variable, $f(z)$ : analytic coefficient, $g(z)$ : analytic coefficient and $w(z)$ : solution Permalink: http://dlmf.nist.gov/1.13.E23 Encodings: TeX, pMML, png See also: Annotations for §1.13(v), §1.13 and Ch.1

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

1.13.24	$\displaystyle\frac{{\mathrm{d}}^{2}U}{{\mathrm{d}z}^{2}}+IU$	$\displaystyle=0,$
1.13.24	$\displaystyle\frac{{\mathrm{d}}^{2}V}{{\mathrm{d}z}^{2}}+JV$	$\displaystyle=0,$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : variable, $U(z)$ : solution, $V(z)$ : solution, $I$ : coefficient and $J$ : coefficient Permalink: http://dlmf.nist.gov/1.13.E24 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §1.13(v), §1.13 and Ch.1

then $W=UV$ is a solution of

1.13.25		$\frac{\mathrm{d}}{\mathrm{d}z}\left(\frac{W^{\prime\prime\prime}+2(I+J)W^{% \prime}+(I^{\prime}+J^{\prime})W}{I-J}\right)=-(I-J)W.$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : variable, $I$ : coefficient, $J$ : coefficient and $W(z)$ : solution Permalink: http://dlmf.nist.gov/1.13.E25 Encodings: TeX, pMML, png See also: Annotations for §1.13(v), §1.13 and Ch.1

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

§1.13(vi) Singularities

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Permalink:: http://dlmf.nist.gov/1.13.vi
See also:: Annotations for §1.13 and Ch.1

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

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Keywords:: closed-form solutions, differential equations
Permalink:: http://dlmf.nist.gov/1.13.vii
See also:: Annotations for §1.13 and Ch.1

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

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Keywords:: Sturm-Liouville form, Sturm-Liouville theory, definition, eigenfunctions, eigenvalue
Referenced by:: Erratum (V1.2.0) §1.13, Version 1.2.0 (March 27, 2024)
Permalink:: http://dlmf.nist.gov/1.13.viii
Addition (effective with 1.2.0):: This subsection was added.
See also:: Annotations for §1.13 and Ch.1

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,
ⓘ Symbols: $u(x)$ : solution, $\alpha$ : variable, $\beta$ : variable, $\alpha^{\prime}$ : variable, $\beta^{\prime}$ : variable, $a$ : real variable and $b$ : real variable Permalink: http://dlmf.nist.gov/1.13.E27 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §1.13(viii), §1.13 and Ch.1

or periodic boundary conditions

1.13.28	$\displaystyle u(a)$	$\displaystyle=u(b),$
1.13.28	$\displaystyle u^{\prime}(a)$	$\displaystyle=u^{\prime}(b),$
ⓘ 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

ⓘ

See also:: Annotations for §1.13(viii), §1.13 and Ch.1

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

ⓘ

See also:: Annotations for §1.13(viii), §1.13 and Ch.1

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

1.13.29		$\ddot{w}(t)+\left(\lambda-\widehat{q}(t)\right)w(t)=0,$
$t\in[0,c]$
ⓘ Symbols: $[\NVar{a},\NVar{b}]$ : closed interval, $\in$ : element of, $\widehat{q}(t)$ : function, $w(t)$ : solution, $\lambda$ : real parameter, $t$ : real variable, $c$ : real variable and $\ddot{\NVar{z}}$ : 2nd derivative of $\NVar{w}$ with respect to $t$ Referenced by: §1.13(viii), §1.18 Permalink: http://dlmf.nist.gov/1.13.E29 Encodings: TeX, pMML, png See also: Annotations for §1.13(viii), §1.13(viii), §1.13 and Ch.1

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

1.13.30	$\displaystyle w(t)$	$\displaystyle=u(x)\left(p(x)\rho(x)\right)^{1/4},$
1.13.30	$\displaystyle t$	$\displaystyle=\int_{a}^{x}\sqrt{\rho{(s)}/p(s)}\,\mathrm{d}s,$
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $p(x)$ : function, $\rho(x)$ : function, $u(x)$ : solution, $w(t)$ : solution, $x$ : real variable, $t$ : real variable and $a$ : real variable Permalink: http://dlmf.nist.gov/1.13.E30 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §1.13(viii), §1.13(viii), §1.13 and Ch.1

and where

1.13.31		$\widehat{q}(t)=q/\rho+\left(p\rho\right)^{-1/4}\frac{{\mathrm{d}}^{2}}{{% \mathrm{d}t}^{2}}\left(p\rho\right)^{1/4}.$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $p(x)$ : function, $q(x)$ : function, $\rho(x)$ : function, $\widehat{q}(t)$ : function and $t$ : real variable Referenced by: §1.13(viii), Erratum (V1.2.0) §1.13 Permalink: http://dlmf.nist.gov/1.13.E31 Encodings: TeX, pMML, png See also: Annotations for §1.13(viii), §1.13(viii), §1.13 and Ch.1

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).

1.12 Continued Fractions 1.14 Integral Transforms

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