List of linear ordinary differential equations
This is a list of named linear ordinary differential equations.
A–Z
Name Order Equation Applications Airy 2 [1] Optics Bessel 2 Wave propagation Cauchy-Euler n Chebyshev 2 Orthogonal polynomials Damped harmonic oscillator 2 Damping Frenet-Serret 1 Differential geometry General Laguerre 2 Hydrogen atom General Legendre 2 Harmonic oscillator 2 Simple harmonic motion Heun 2 Hill 2 , (f periodic) Physics Hypergeometric 2 Kummer 2 Laguerre 2 Legendre 2 Orthogonal polynomials Matrix 1 Picard-Fuchs 2 Elliptic curves Riemann 2 Quantum harmonic oscillator 2 Quantum mechanics Sturm-Liouville 2 Applied mathematics
![{\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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b09b45c5e9c45a07572ce17ac503065c93c68040)
![{\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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4462354d74b707479825649b6b99f492a75ead12)
![{\displaystyle z(1-z){\frac {d^{2}w}{dz^{2}}}+\left[c-(a+b+1)z\right]{\frac {dw}{dz}}-ab\,w=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/31043b523d4dd446913f3838f1ddb26ba9bef733)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff4756b2afc63eed01a52aaace0af8a70828204d)
![{\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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/edd0b3392ef2f62214eac8d4b1c54c285d45912d)
![{\displaystyle {\frac {d}{dx}}\!\!\left[\,p(x){\frac {dy}{dx}}\right]+q(x)y=-\lambda \,w(x)y,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b7ed22ff2571ca9f5bc6f1236c5f57aea09ec786)
See also
- List of nonlinear ordinary differential equations
- List of nonlinear partial differential equations
- List of named differential equations
References
- ^ Vallée, Olivier (2004). Airy functions and applications to physics. Internet Archive. Hackensack, NJ : World Scientific. p. 5. ISBN 978-1-86094-478-9.