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This is a list of named linear ordinary differential equations .[ 1]
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
^ [keepnotes.com/mit/multivariable-calculus/100-linear-approximations-and-tangent-planes-linear-approximation-review "Linear Approximations and Tangent Planes"]. KeepNotes . Massachusetts Institute of Technology.