| Name |
Order |
Equation |
Applications |
| Airy |
2 |
 |
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 |
![{\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}](//wikimedia.org/api/rest_v1/media/math/render/svg/b09b45c5e9c45a07572ce17ac503065c93c68040) |
|
| Harmonic oscillator |
2 |
 |
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}](//wikimedia.org/api/rest_v1/media/math/render/svg/4462354d74b707479825649b6b99f492a75ead12) |
|
| Hill |
2 |
 , (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}](//wikimedia.org/api/rest_v1/media/math/render/svg/31043b523d4dd446913f3838f1ddb26ba9bef733) |
|
| Kummer |
2 |
 |
|
| Laguerre |
2 |
 |
|
| Legendre |
2 |
 |
Orthogonal polynomials |
| Matrix |
1 |
 |
|
| Picard-Fuchs |
2 |
 |
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}}}](//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}](//wikimedia.org/api/rest_v1/media/math/render/svg/edd0b3392ef2f62214eac8d4b1c54c285d45912d)
|
|
| Quantum harmonic oscillator |
2 |
 |
Quantum mechanics |
| Sturm-Liouville |
2 |
![{\displaystyle {\frac {d}{dx}}\!\!\left[\,p(x){\frac {dy}{dx}}\right]+q(x)y=-\lambda \,w(x)y,}](//wikimedia.org/api/rest_v1/media/math/render/svg/b7ed22ff2571ca9f5bc6f1236c5f57aea09ec786) |
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}](http://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}](http://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}](http://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}}}](http://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}](http://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,}](http://wikimedia.org/api/rest_v1/media/math/render/svg/b7ed22ff2571ca9f5bc6f1236c5f57aea09ec786)