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 [math]\displaystyle{ \frac{d^2y}{dx^2} - xy = 0 }[/math] Optics Bessel 2 [math]\displaystyle{ x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + \left(x^2 - \alpha^2 \right)y = 0 }[/math] Wave propagation Cauchy-Euler n [math]\displaystyle{ a_{n} x^n y^{(n)}(x) + a_{n-1} x^{n-1} y^{(n-1)}(x) + \dots + a_0 y(x) = 0 }[/math] Chebyshev 2 [math]\displaystyle{ (1 - x^2)y'' - xy' + n^2 y = 0,\quad(1 - x^2)y'' - 3xy' + n(n + 2) y = 0 }[/math] Orthogonal polynomials Damped harmonic oscillator 2 [math]\displaystyle{ m \frac{\mathrm{d}^2x}{\mathrm{d}t^2}+ c\frac{\mathrm{d}x}{\mathrm{d}t} +kx =0 }[/math] Damping Frenet-Serret 1 [math]\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} }[/math] Differential geometry General Laguerre 2 [math]\displaystyle{ xy'' + (\alpha + 1 - x)y' + ny = 0 }[/math] Hydrogen atom General Legendre 2 [math]\displaystyle{ \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 }[/math] Harmonic oscillator 2 [math]\displaystyle{ m \frac{\mathrm{d}^2x}{\mathrm{d}t^2} +k x =0 }[/math] Simple harmonic motion Heun 2 [math]\displaystyle{ \frac {d^2w}{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 }[/math] Hill 2 [math]\displaystyle{ \frac{d^2y}{dt^2} + f(t) y = 0 }[/math], (f periodic) Physics Hypergeometric 2 [math]\displaystyle{ z(1-z)\frac {d^2w}{dz^2} + \left[c-(a+b+1)z \right] \frac {dw}{dz} - ab\,w = 0 }[/math] Kummer 2 [math]\displaystyle{ z\frac{d^2w}{dz^2} + (b-z)\frac{dw}{dz} - aw = 0 }[/math] Laguerre 2 [math]\displaystyle{ xy'' + (1 - x)y' + ny = 0 }[/math] Legendre 2 [math]\displaystyle{ (1 - x^2) P_n''(x) - 2 x P_n'(x) + n (n + 1) P_n(x) = 0 }[/math] Orthogonal polynomials Matrix 1 [math]\displaystyle{ \mathbf{\dot{x}}(t) = \mathbf{A}(t)\mathbf{x}(t) }[/math] Picard-Fuchs 2 [math]\displaystyle{ \frac{d^2y}{dj^2} + \frac{1}{j} \frac{dy}{dj} + \frac{31j -4}{144j^2(1-j)^2} y=0 }[/math] Elliptic curves Riemann 2 [math]\displaystyle{ \frac{d^2w}{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} }[/math] - [math]\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 }[/math]
Quantum harmonic oscillator 2 [math]\displaystyle{ -\frac{1}{2}\frac{d^2\psi}{dx^2} + \frac{1}{2} x^2 \psi = E\psi }[/math] Quantum mechanics Sturm-Liouville 2 [math]\displaystyle{ \frac{d}{dx}\!\!\left[\,p(x)\frac{dy}{dx}\right] + q(x)y = -\lambda\, w(x)y, }[/math] 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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