List of linear ordinary differential equations

This is a list of named linear ordinary differential equations.

A–Z

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

See also

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