differential equation

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1: 30.2 Differential Equations

§30.2 Differential Equations

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§30.2(i) Spheroidal Differential Equation

… ► … ►The Liouville normal form of equation (30.2.1) is … ►

§30.2(iii) Special Cases

…

2: 15.10 Hypergeometric Differential Equation

§15.10 Hypergeometric Differential Equation

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§15.10(i) Fundamental Solutions

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15.10.1 $$z(1-z)\frac{d^{2}w}{{dz}^2}+\left(c-(a+b+1)z\right)\frac{dw}{dz}-abw=0.$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Referenced by:

§15.11(i), §15.11(i), §15.17(ii), §15.19(ii), §15.2(ii), §15.5(ii), §16.13, §17.6(iv), §19.18(ii), §19.4(ii)

Permalink:

http://dlmf.nist.gov/15.10.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §15.10(i), §15.10 and Ch.15

►This is the hypergeometric differential equation. … ► …

3: 16.25 Methods of Computation

… ►Methods for computing the functions of the present chapter include power series, asymptotic expansions, integral representations, differential equations, and recurrence relations. …There is, however, an added feature in the numerical solution of differential equations and difference equations (recurrence relations). …

4: 10.72 Mathematical Applications

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§10.72(i) Differential Equations with Turning Points

… ►The canonical form of differential equation for these problems is given by … ►

§10.72(ii) Differential Equations with Poles

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§10.72(iii) Differential Equations with a Double Pole and a Movable Turning Point

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5: 9.15 Mathematical Applications

… ►Airy functions play an indispensable role in the construction of uniform asymptotic expansions for contour integrals with coalescing saddle points, and for solutions of linear second-order ordinary differential equations with a simple turning point. …

6: 10.36 Other Differential Equations

§10.36 Other Differential Equations

… ►Differential equations for products can be obtained from (10.13.9)–(10.13.11) by replacing $z$ by $\mathrm{i}z$.

7: 18.8 Differential Equations

§18.8 Differential Equations

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Table 18.8.1: Classical OP’s: differential equations $A(x)f^{\prime \prime }(x)+B(x)f^{\prime }(x)+C(x)f(x)+{\lambda }_{n}f(x)=0$.

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#	$f(x)$	$A(x)$	$B(x)$	$C(x)$	${\lambda }_n$
…

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8: 15.11 Riemann’s Differential Equation

§15.11 Riemann’s Differential Equation

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§15.11(i) Equations with Three Singularities

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§15.11(ii) Transformation Formulas

… ►The reduction of a general homogeneous linear differential equation of the second order with at most three regular singularities to the hypergeometric differential equation is given by … ►for arbitrary $\lambda$ and $\mu$.

9: 3.7 Ordinary Differential Equations

§3.7 Ordinary Differential Equations

… ► … ► … ►

§3.7(iii) Taylor-Series Method: Boundary-Value Problems

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§3.7(v) Runge–Kutta Method

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10: 13.27 Mathematical Applications

… ►For applications of Whittaker functions to the uniform asymptotic theory of differential equations with a coalescing turning point and simple pole see §§2.8(vi) and 18.15(i).