List of linear ordinary differential equations explained

This is a list of named linear ordinary differential equations.

A–Z

Name	Order	Equation	Applications
Airy	2	(d ^{2y})/(dx ^{2}) - xy = 0	Optics
Bessel	2	x ^{2} (d ^{2} y)/(dx ^{2}) + x (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) + ... + a _{0} y(x) = 0
Chebyshev	2	( 1 - x ^{2)y''} - xy' + n ^{2} y = 0 , ( 1 - x ^{2)y''} - 3 xy' + n(n + 2 ) y = 0	Orthogonal polynomials
Damped harmonic oscillator	2	m (d ^{2x})/(d t ^{2}) + c (d x)/(d t) + kx =0	Damping
Frenet-Serret	1	\dfrac{ d T }{ d s } = \kappa N , \dfrac{ d N }{ d s } = - \kappa T + \tau B , \dfrac{ d B }{ d s } = - \tau N	Differential geometry
General Laguerre	2	xy'' + (\alpha + 1 - x)y' + ny = 0	Hydrogen atom
General Legendre	2	\left( 1 - x ^{2\right)} (d ^{2})/(d x ^{2}) P ^{m(x)} _{\ell} - 2 x (d)/(d x) P ^{m(x)} _{\ell} + \left[ \ell (\ell + 1 ) - (m ^{2})/(1- x ^{2}) \right] P ^{m(x)} _{\ell} = 0
Harmonic oscillator	2	m (d ^{2x})/(d t ^{2}) + k x =0	Simple harmonic motion
Heun	2	(d ^{2w})/(dz ^{2}) + \left[ (\gamma)/(z) + (\delta)/(z -1) + (\epsilon)/(z - a) \right] (dw)/(dz) + (\alpha \beta z - q)/(z(z -1 )(z - a)) w = 0
Hill	2	(d ^{2y})/(dt ^{2}) + f(t) y = 0 , (f periodic)	Physics
Hypergeometric	2	z( 1- z) (d ^{2w})/(dz ^{2}) + \left[c - (a + b +1 )z \right] (dw)/(dz) - abw = 0
Kummer	2	z (d ^{2w})/(dz ^{2}) + (b - z) (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	• x (t) = A (t) x (t)
Picard-Fuchs	2	(d ^{2y})/(dj ^{2}) + (1)/(j) (dy)/(dj) + (31 j -4)/(144 j ^{2(1-j)} ^{2}) y =0	Elliptic curves
Riemann	2	(d ^{2w})/(dz ^{2}) + \left[ (1- \alpha - \alpha')/(+ z - a) (1- \beta - \beta')/(+ z - b) (1- \gamma - \gamma')/(z - c) \right] (dw)/(dz) + \left[ (\alpha\alpha' (a - b)(a - c))/(+ z - a) (\beta\beta' (b - c)(b - a))/(+ z - b) (\gamma\gamma' (c - a)(c - b))/(\right] z - c) (w)/((z - a)(z - b)(z - c)) =0
Quantum harmonic oscillator	2	- (1)/(2) (d ^{2\psi})/(dx ^{2}) + (1)/(2) x ^{2} \psi = E\psi	Quantum mechanics
Sturm-Liouville	2	(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