List of linear ordinary differential equations

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This is a list of named linear ordinary differential equations.

A–Z

Name	Order	Equation	Applications
Airy	2	$\frac{d^{2}y}{d{x}^2}-xy=0$	Optics
Bessel	2	$x^2\frac{d^{2}y}{d{x}^2}+x\frac{dy}{dx}+(x^2-{\alpha }^2)y=0$	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	$(1-x^2)y^{″}-x{y}^{\prime }+n^{2}y=0,(1-x^2)y^{″}-3x{y}^{\prime }+n(n+2)y=0$	Orthogonal polynomials
Damped harmonic oscillator	2	$m\frac{{\mathrm{d}}^{2}x}{\mathrm{d}{t}^2}+c\frac{\mathrm{d}x}{\mathrm{d}t}+kx=0$	Damping
Frenet-Serret	1	$\frac{\mathrm{d}\mathbf{𝐓}}{\mathrm{d}s}=\kappa \mathbf{𝐍},\frac{\mathrm{d}\mathbf{𝐍}}{\mathrm{d}s}=-\kappa \mathbf{𝐓}+\tau \mathbf{𝐁},\frac{\mathrm{d}\mathbf{𝐁}}{\mathrm{d}s}=-\tau \mathbf{𝐍}$	Differential geometry
General Laguerre	2	$x{y}^{″}+(\alpha +1-x)y^{\prime }+ny=0$	Hydrogen atom
General Legendre	2	$(1-x^2)\frac{d^2}{d{x}^2}{P}_{\ell }^m(x)-2x\frac{d}{dx}{P}_{\ell }^m(x)+[\ell (\ell +1)-\frac{m^2}{1-x^2}]P_{\ell }^m(x)=0$
Harmonic oscillator	2	$m\frac{{\mathrm{d}}^{2}x}{\mathrm{d}{t}^2}+kx=0$	Simple harmonic motion
Heun	2	$\frac{d^{2}w}{d{z}^2}+[\frac{\gamma }{z}+\frac{\delta }{z-1}+\frac{ϵ}{z-a}]\frac{dw}{dz}+\frac{\alpha \beta z-q}{z(z-1)(z-a)}w=0$
Hill	2	$\frac{d^{2}y}{d{t}^2}+f(t)y=0$, (f periodic)	Physics
Hypergeometric	2	$z(1-z)\frac{d^{2}w}{d{z}^2}+[c-(a+b+1)z]\frac{dw}{dz}-abw=0$
Kummer	2	$z\frac{d^{2}w}{d{z}^2}+(b-z)\frac{dw}{dz}-aw=0$
Laguerre	2	$x{y}^{″}+(1-x)y^{\prime }+ny=0$
Legendre	2	$(1-x^2){P_n}^{″}(x)-2x{P_n}^{\prime }(x)+n(n+1)P_n(x)=0$	Orthogonal polynomials
Matrix	1	$\dot{\mathbf{x}}(t)=\mathbf{𝐀}(t)\mathbf{𝐱}(t)$
Picard-Fuchs	2	$\frac{d^{2}y}{d{j}^2}+\frac{1}{j}\frac{dy}{dj}+\frac{31j-4}{144j^2(1-j{)}^2}y=0$	Elliptic curves
Riemann	2	$\frac{d^{2}w}{d{z}^2}+[\frac{1-\alpha -{\alpha }^{\prime }}{z-a}+\frac{1-\beta -{\beta }^{\prime }}{z-b}+\frac{1-\gamma -{\gamma }^{\prime }}{z-c}]\frac{dw}{dz}$ $+[\frac{\alpha {\alpha }^{\prime }(a-b)(a-c)}{z-a}+\frac{\beta {\beta }^{\prime }(b-c)(b-a)}{z-b}+\frac{\gamma {\gamma }^{\prime }(c-a)(c-b)}{z-c}]\frac{w}{(z-a)(z-b)(z-c)}=0$
Quantum harmonic oscillator	2	$-\frac{1}{2}\frac{d^2\psi }{d{x}^2}+\frac{1}{2}{x}^2\psi =E\psi$	Quantum mechanics
Sturm-Liouville	2	$\frac{d}{dx}[p(x)\frac{dy}{dx}]+q(x)y=-\lambda w(x)y,$	Applied mathematics

See also

• List of nonlinear ordinary differential equations
• List of nonlinear partial differential equations
• List of named differential equations