List of linear ordinary differential equations

This is a list of named linear ordinary differential equations.

A–Z

Name Order Equation Applications
Airy 2 d 2 y d x 2 − x y = 0 {\displaystyle {\frac {d^{2}y}{dx^{2}}}-xy=0} Optics
Bessel 2 x 2 d 2 y d x 2 + x d y d x + ( x 2 − α 2 ) y = 0 {\displaystyle x^{2}{\frac {d^{2}y}{dx^{2}}}+x{\frac {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 {\displaystyle a_{n}x^{n}y^{(n)}(x)+a_{n-1}x^{n-1}y^{(n-1)}(x)+\dots +a_{0}y(x)=0}
Chebyshev 2 ( 1 − x 2 ) y ″ − x y ′ + n 2 y = 0 , ( 1 − x 2 ) y ″ − 3 x y ′ + n ( n + 2 ) y = 0 {\displaystyle (1-x^{2})y''-xy'+n^{2}y=0,\quad (1-x^{2})y''-3xy'+n(n+2)y=0} Orthogonal polynomials
Damped harmonic oscillator 2 m d 2 x d t 2 + c d x d t + k x = 0 {\displaystyle m{\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}+c{\frac {\mathrm {d} x}{\mathrm {d} t}}+kx=0} Damping
Frenet-Serret 1 d T d s = κ N , d N d s = − κ T + τ B , d B d s = − τ N {\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} } Differential geometry
General Laguerre 2 x y ″ + ( α + 1 − x ) y ′ + n y = 0 {\displaystyle xy''+(\alpha +1-x)y'+ny=0} Hydrogen atom
General Legendre 2 ( 1 − x 2 ) d 2 d x 2 P ℓ m ( x ) − 2 x d d x P ℓ m ( x ) + [ ℓ ( ℓ + 1 ) − m 2 1 − x 2 ] P ℓ m ( x ) = 0 {\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}
Harmonic oscillator 2 m d 2 x d t 2 + k x = 0 {\displaystyle m{\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}+kx=0} Simple harmonic motion
Heun 2 d 2 w d z 2 + [ γ z + δ z − 1 + ϵ z − a ] d w d z + α β z − q z ( z − 1 ) ( z − a ) w = 0 {\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}
Hill 2 d 2 y d t 2 + f ( t ) y = 0 {\displaystyle {\frac {d^{2}y}{dt^{2}}}+f(t)y=0} , (f periodic) Physics
Hypergeometric 2 z ( 1 − z ) d 2 w d z 2 + [ c − ( a + b + 1 ) z ] d w d z − a b w = 0 {\displaystyle z(1-z){\frac {d^{2}w}{dz^{2}}}+\left[c-(a+b+1)z\right]{\frac {dw}{dz}}-ab\,w=0}
Kummer 2 z d 2 w d z 2 + ( b − z ) d w d z − a w = 0 {\displaystyle z{\frac {d^{2}w}{dz^{2}}}+(b-z){\frac {dw}{dz}}-aw=0}
Laguerre 2 x y ″ + ( 1 − x ) y ′ + n y = 0 {\displaystyle 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 {\displaystyle (1-x^{2})P_{n}''(x)-2xP_{n}'(x)+n(n+1)P_{n}(x)=0} Orthogonal polynomials
Matrix 1 x ˙ ( t ) = A ( t ) x ( t ) {\displaystyle \mathbf {\dot {x}} (t)=\mathbf {A} (t)\mathbf {x} (t)}
Picard-Fuchs 2 d 2 y d j 2 + 1 j d y d j + 31 j − 4 144 j 2 ( 1 − j ) 2 y = 0 {\displaystyle {\frac {d^{2}y}{dj^{2}}}+{\frac {1}{j}}{\frac {dy}{dj}}+{\frac {31j-4}{144j^{2}(1-j)^{2}}}y=0} Elliptic curves
Riemann 2 d 2 w d z 2 + [ 1 − α − α ′ z − a + 1 − β − β ′ z − b + 1 − γ − γ ′ z − c ] d w d z {\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}}}
+ [ α α ′ ( a − b ) ( a − c ) z − a + β β ′ ( b − c ) ( b − a ) z − b + γ γ ′ ( c − a ) ( c − b ) z − c ] w ( z − a ) ( z − b ) ( z − c ) = 0 {\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}
Quantum harmonic oscillator 2 − 1 2 d 2 ψ d x 2 + 1 2 x 2 ψ = E ψ {\displaystyle -{\frac {1}{2}}{\frac {d^{2}\psi }{dx^{2}}}+{\frac {1}{2}}x^{2}\psi =E\psi } Quantum mechanics
Sturm-Liouville 2 d d x [ p ( x ) d y d x ] + q ( x ) y = − λ w ( x ) y , {\displaystyle {\frac {d}{dx}}\!\!\left[\,p(x){\frac {dy}{dx}}\right]+q(x)y=-\lambda \,w(x)y,} Applied mathematics

See also