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

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Airy | 2  | d2ydx2−xy=0 | Optics
Bessel | 2  | x2d2ydx2+xdydx+(x2−α2)y=0 | Wave propagation
Cauchy-Euler  | n  | anxny(n)(x)+an−1xn−1y(n−1)(x)+…+a0y(x)=0 |
Chebyshev | 2  | (1−x2)y″−xy′+n2y=0,(1−x2)y″−3xy′+n(n+2)y=0 | Orthogonal polynomials
Damped harmonic oscillator | 2  | md2xdt2+cdxdt+kx=0 | Damping
Frenet-Serret  | 1  | d𝐓ds=κ𝐍,d𝐍ds=−κ𝐓+τ𝐁,d𝐁ds=−τ𝐍 | Differential geometry
General Laguerre | 2  | xy″+(α+1−x)y′+ny=0 | Hydrogen atom
General Legendre | 2  | (1−x2)d2dx2Pℓm(x)−2xddxPℓm(x)+[ℓ(ℓ+1)−m21−x2]Pℓm(x)=0 |
Harmonic oscillator | 2  | md2xdt2+kx=0 | Simple harmonic motion
Heun | 2  | d2wdz2+[γz+δz−1+ϵz−a]dwdz+αβz−qz(z−1)(z−a)w=0 |
Hill | 2  | d2ydt2+f(t)y=0, (f periodic)  | Physics
Hypergeometric | 2  | z(1−z)d2wdz2+[c−(a+b+1)z]dwdz−abw=0 |
Kummer | 2  | zd2wdz2+(b−z)dwdz−aw=0 |
Laguerre | 2  | xy″+(1−x)y′+ny=0 |
Legendre | 2  | (1−x2)Pn″(x)−2xPn′(x)+n(n+1)Pn(x)=0 | Orthogonal polynomials
Matrix | 1  | x˙(t)=𝐀(t)𝐱(t) |
Picard-Fuchs  | 2  | d2ydj2+1jdydj+31j−4144j2(1−j)2y=0 | Elliptic curves
Riemann | 2  | d2wdz2+[1−α−α′z−a+1−β−β′z−b+1−γ−γ′z−c]dwdz

+[αα′(a−b)(a−c)z−a+ββ′(b−c)(b−a)z−b+γγ′(c−a)(c−b)z−c]w(z−a)(z−b)(z−c)=0

Quantum harmonic oscillator | 2  | −12d2ψdx2+12x2ψ=Eψ | Quantum mechanics
Sturm-Liouville  | 2  | ddx[p(x)dydx]+q(x)y=−λw(x)y, | Applied mathematics

## See also

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

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