# Euler?Poisson differential equations

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##### 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: 31.2 Differential Equations
###### §31.2(i) Heun’s Equation
31.2.1 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\left(\frac{\gamma}{z}+\frac{% \delta}{z-1}+\frac{\epsilon}{z-a}\right)\frac{\mathrm{d}w}{\mathrm{d}z}+\frac{% \alpha\beta z-q}{z(z-1)(z-a)}w=0,$ $\alpha+\beta+1=\gamma+\delta+\epsilon$.
All other homogeneous linear differential equations of the second order having four regular singularities in the extended complex plane, $\mathbb{C}\cup\{\infty\}$, can be transformed into (31.2.1). …
##### 4: 29.2 Differential Equations
###### §29.2(ii) Other Forms
Equation (29.2.10) is a special case of Heun’s equation (31.2.1).
##### 5: 32.2 Differential Equations
###### §32.2(i) Introduction
The six Painlevé equations $\mbox{P}_{\mbox{\scriptsize I}}$$\mbox{P}_{\mbox{\scriptsize VI}}$ are as follows: … be a nonlinear second-order differential equation in which $F$ is a rational function of $w$ and $\ifrac{\mathrm{d}w}{\mathrm{d}z}$, and is locally analytic in $z$, that is, analytic except for isolated singularities in $\mathbb{C}$. …
##### 6: 28.2 Definitions and Basic Properties
###### §28.2(i) Mathieu’s Equation
This is the characteristic equation of Mathieu’s equation (28.2.1). …
##### 7: 28.20 Definitions and Basic Properties
###### §28.20(i) Modified Mathieu’s Equation
When $z$ is replaced by $\pm\mathrm{i}z$, (28.2.1) becomes the modified Mathieu’s equation:
28.20.8 $h=\sqrt{q}\;(>0).$
Then from §2.7(ii) it is seen that equation (28.20.2) has independent and unique solutions that are asymptotic to $\zeta^{\ifrac{1}{2}}e^{\pm 2\mathrm{i}h\zeta}$ as $\zeta\to\infty$ in the respective sectors $|\operatorname{ph}\left(\mp\mathrm{i}\zeta\right)|\leq\tfrac{3}{2}\pi-\delta$, $\delta$ being an arbitrary small positive constant. …
##### 8: 19.18 Derivatives and Differential Equations
###### §19.18(ii) DifferentialEquations
and also a system of $n(n-1)/2$ Euler–Poisson differential equations (of which only $n-1$ are independent): …If $n=2$, then elimination of $\partial_{2}v$ between (19.18.11) and (19.18.12), followed by the substitution $(b_{1},b_{2},z_{1},z_{2})=(b,c-b,1-z,1)$, produces the Gauss hypergeometric equation (15.10.1). The next four differential equations apply to the complete case of $R_{F}$ and $R_{G}$ in the form $R_{-a}\left(\tfrac{1}{2},\tfrac{1}{2};z_{1},z_{2}\right)$ (see (19.16.20) and (19.16.23)). …
##### 9: 1.5 Calculus of Two or More Variables
The function $f(x,y)$ is continuously differentiable if $f$, $\ifrac{\partial f}{\partial x}$, and $\ifrac{\partial f}{\partial y}$ are continuous, and twice-continuously differentiable if also $\ifrac{{\partial}^{2}f}{{\partial x}^{2}}$, $\ifrac{{\partial}^{2}f}{{\partial y}^{2}}$, $\,{\partial}^{2}f/\,\partial x\,\partial y$, and $\,{\partial}^{2}f/\,\partial y\,\partial x$ are continuous. … If $F(x,y)$ is continuously differentiable, $F(a,b)=0$, and $\ifrac{\partial F}{\partial y}\not=0$ at $(a,b)$, then in a neighborhood of $(a,b)$, that is, an open disk centered at $a,b$, the equation $F(x,y)=0$ defines a continuously differentiable function $y=g(x)$ such that $F(x,g(x))=0$, $b=g(a)$, and $g^{\prime}(x)=-F_{x}/F_{y}$. … Equations (1.5.11) and (1.5.12) still apply, but … Suppose that $a,b,c$ are finite, $d$ is finite or $+\infty$, and $f(x,y)$, $\ifrac{\partial f}{\partial x}$ are continuous on the partly-closed rectangle or infinite strip $[a,b]\times[c,d)$. Suppose also that $\int^{d}_{c}f(x,y)\,\mathrm{d}y$ converges and $\int^{d}_{c}(\ifrac{\partial f}{\partial x})\,\mathrm{d}y$ converges uniformly on $a\leq x\leq b$, that is, given any positive number $\epsilon$, however small, we can find a number $c_{0}\in[c,d)$ that is independent of $x$ and is such that …
##### 10: 36.10 Differential Equations
###### §36.10 DifferentialEquations
$K=2$, cusp: … $K=3$, swallowtail: … In terms of the normal forms (36.2.2) and (36.2.3), the $\Psi^{(\mathrm{U})}\left(\mathbf{x}\right)$ satisfy the following operator equationsEquation (36.10.17) is the paraxial wave equation. …