# differential equations

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## 1—10 of 236 matching pages

##### 1: 30.2 Differential Equations
###### §30.2(i) Spheroidal DifferentialEquation
The Liouville normal form of equation (30.2.1) is …
##### 2: 15.10 Hypergeometric Differential Equation
###### §15.10(i) Fundamental Solutions
15.10.1 $z(1-z)\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\left(c-(a+b+1)z\right)\frac% {\mathrm{d}w}{\mathrm{d}z}-abw=0.$
This is the hypergeometric differential equation. …
##### 3: 30.3 Eigenvalues
###### §30.3 Eigenvalues
With $\mu=m=0,1,2,\dots$, the spheroidal wave functions $\mathsf{Ps}^{m}_{n}\left(x,\gamma^{2}\right)$ are solutions of Equation (30.2.1) which are bounded on $(-1,1)$, or equivalently, which are of the form $(1-x^{2})^{\frac{1}{2}m}g(x)$ where $g(z)$ is an entire function of $z$. …
###### §30.3(iv) Power-Series Expansion
Further coefficients can be found with the Maple program SWF9; see §30.18(i).
##### 4: 15.11 Riemann’s Differential Equation
###### §15.11(i) Equations with Three Singularities
15.11.3 $w=P\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}.$
###### §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 …
##### 5: 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). …
##### 7: 10.72 Mathematical Applications
###### §10.72(i) DifferentialEquations with Turning Points
The canonical form of differential equation for these problems is given by …
##### 8: 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. …
##### 9: 10.36 Other Differential Equations
###### §10.36 Other DifferentialEquations
Differential equations for products can be obtained from (10.13.9)–(10.13.11) by replacing $z$ by $iz$.