The_Cambridge_Handbook_of_Physics_Formulas

2.7 Differentiation
43
Differential equations
Laplace ∇ 2 f = 0 (2.339) f f(x,y,z)
Diffusion a
∂f
∂t = D∇2 f (2.340)
D
diffusion
coefficient
2
Helmholtz ∇ 2 f +α 2 f = 0 (2.341) α constant
Wave
Legendre
Associated
Legendre
Bessel
Hermite
Laguerre
Associated
Laguerre
Chebyshev
Euler (or
Cauchy)
Bernoulli
∇ 2 f = 1 ∂ 2 f
c 2 ∂t 2 (2.342) c wave speed
[
d
(1−x 2 ) dy ]
+l(l +1)y = 0
dx dx
(2.343) l integer
d
dx
[
(1−x 2 ) dy
dx
]
]
+
[l(l +1)− m2
1−x 2 y = 0 (2.344) m integer
x 2 d2 y dy
+x
dx2 dx +(x2 −m 2 )y = 0 (2.345)
d 2 y dy
−2x +2αy = 0 (2.346)
dx2 dx
x d2 y dy
+(1−x) +αy = 0 (2.347)
dx2 dx
x d2 y
dy
+(1+k −x)
dx2 dx +αy = 0 (2.348) k integer
(1−x 2 ) d2 y dy
−x
dx2 dx +n2 y = 0 (2.349) n integer
x 2 d2 y dy
+ax
dx2 dx +by = f(x) (2.350) a,b constants
dy
dx +p(x)y = q(x)ya (2.351) p,q functions of x
d
Airy
2 y
= xy (2.352)
dx2 a Also known as the “conduction equation.” For thermal conduction, f ≡ T and D, the thermal diffusivity,
≡ κ ≡ λ/(ρc p ), where T is the temperature distribution, λ the thermal conductivity, ρ the density, and c p the specific
heat capacity of the material.