List of linear ordinary differential equations

This is a list of named linear ordinary differential equations.

A–Z

 * {| class="wikitable sortable" style="background: white; color: black; text-align: left"

!Name !Order !Equation !Applications \dfrac{ \mathrm{d} \mathbf{T} }{ \mathrm{d} s } =\kappa \mathbf{N},\quad \dfrac{ \mathrm{d} \mathbf{N} }{ \mathrm{d} s } = -\kappa \mathbf{T} +\, \tau \mathbf{B},\quad \dfrac{ \mathrm{d} \mathbf{B} }{ \mathrm{d} s } = -\tau \mathbf{N} $$ \left[\frac{\gamma}{z}+ \frac{\delta}{z-1} + \frac{\epsilon}{z-a} \right] \frac {dw}{dz} + \frac {\alpha \beta z -q} {z(z-1)(z-a)} w = 0$$ \frac{31j -4}{144j^2(1-j)^2} y=0$$ \frac{1-\alpha-\alpha'}{z-a} + \frac{1-\beta-\beta'}{z-b} + \frac{1-\gamma-\gamma'}{z-c} \right] \frac{dw}{dz} $$
 * -style="background: #eee"
 * Airy
 * 2
 * $$\frac{d^2y}{dx^2} - xy = 0$$
 * Optics
 * Bessel
 * 2
 * $$x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + \left(x^2 - \alpha^2 \right)y = 0$$
 * Wave propagation
 * Cauchy-Euler
 * n
 * $$a_{n} x^n y^{(n)}(x) + a_{n-1} x^{n-1} y^{(n-1)}(x) + \dots + a_0 y(x) = 0$$
 * Chebyshev
 * 2
 * $$(1 - x^2)y - xy' + n^2 y = 0,\quad(1 - x^2)y - 3xy' + n(n + 2) y = 0$$
 * Orthogonal polynomials
 * Damped harmonic oscillator
 * 2
 * $$ m \frac{\mathrm{d}^2x}{\mathrm{d}t^2}+  c\frac{\mathrm{d}x}{\mathrm{d}t} +kx  =0$$
 * Damping
 * Frenet-Serret
 * 1
 * Damped harmonic oscillator
 * 2
 * $$ m \frac{\mathrm{d}^2x}{\mathrm{d}t^2}+  c\frac{\mathrm{d}x}{\mathrm{d}t} +kx  =0$$
 * Damping
 * Frenet-Serret
 * 1
 * 1
 * Differential geometry
 * General Laguerre
 * 2
 * $$xy'' + (\alpha + 1 - x)y' + ny = 0$$
 * Hydrogen atom
 * General Legendre
 * 2
 * $$\left(1 - x^2\right) \frac{d^2}{d x^2} P_\ell^m(x) - 2 x \frac{d}{d x} P_\ell^m(x) + \left[ \ell (\ell + 1) - \frac{m^2}{1 - x^2} \right] P_\ell^m(x) = 0$$
 * Harmonic oscillator
 * 2
 * $$m \frac{\mathrm{d}^2x}{\mathrm{d}t^2} +k x =0$$
 * Simple harmonic motion
 * Heun
 * 2
 * $$\frac {d^2w}{dz^2} +
 * Simple harmonic motion
 * Heun
 * 2
 * $$\frac {d^2w}{dz^2} +
 * $$\frac {d^2w}{dz^2} +
 * Hill
 * 2
 * $$ \frac{d^2y}{dt^2} + f(t) y = 0$$, (f periodic)
 * Physics
 * Hypergeometric
 * 2
 * $$z(1-z)\frac {d^2w}{dz^2} + \left[c-(a+b+1)z \right] \frac {dw}{dz} - ab\,w = 0$$
 * Kummer
 * 2
 * $$z\frac{d^2w}{dz^2} + (b-z)\frac{dw}{dz} - aw = 0$$
 * Laguerre
 * 2
 * $$xy'' + (1 - x)y' + ny = 0$$
 * Legendre
 * 2
 * $$(1 - x^2) P_n''(x) - 2 x P_n'(x) + n (n + 1) P_n(x) = 0$$
 * Orthogonal polynomials
 * Matrix
 * 1
 * $$\mathbf{\dot{x}}(t) = \mathbf{A}(t)\mathbf{x}(t)$$
 * Picard-Fuchs
 * 2
 * $$\frac{d^2y}{dj^2} + \frac{1}{j} \frac{dy}{dj} +
 * 2
 * $$(1 - x^2) P_n''(x) - 2 x P_n'(x) + n (n + 1) P_n(x) = 0$$
 * Orthogonal polynomials
 * Matrix
 * 1
 * $$\mathbf{\dot{x}}(t) = \mathbf{A}(t)\mathbf{x}(t)$$
 * Picard-Fuchs
 * 2
 * $$\frac{d^2y}{dj^2} + \frac{1}{j} \frac{dy}{dj} +
 * Picard-Fuchs
 * 2
 * $$\frac{d^2y}{dj^2} + \frac{1}{j} \frac{dy}{dj} +
 * Elliptic curves
 * Riemann
 * 2
 * $$\frac{d^2w}{dz^2} + \left[
 * $$\frac{d^2w}{dz^2} + \left[
 * $$+\left[

\frac{\alpha\alpha' (a-b)(a-c)} {z-a} +\frac{\beta\beta' (b-c)(b-a)} {z-b} +\frac{\gamma\gamma' (c-a)(c-b)} {z-c} \right] \frac{w}{(z-a)(z-b)(z-c)}=0$$
 * Quantum harmonic oscillator
 * 2
 * $$-\frac{1}{2}\frac{d^2\psi}{dx^2} + \frac{1}{2} x^2 \psi = E\psi $$
 * Quantum mechanics
 * Sturm-Liouville
 * 2
 * $$\frac{d}{dx}\!\!\left[\,p(x)\frac{dy}{dx}\right] + q(x)y = -\lambda\, w(x)y, $$
 * Applied mathematics
 * }
 * $$\frac{d}{dx}\!\!\left[\,p(x)\frac{dy}{dx}\right] + q(x)y = -\lambda\, w(x)y, $$
 * Applied mathematics
 * }