Laguerre Differential Equation

Author

Eric W. Weisstein
October 5, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/DifferentialEquations/LaguerreDifferentialEquation.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/LaguerreDifferentialEquation.html.

©2008 Wolfram Research, Inc. except for portions noted otherwise

Non-Associative

Equation

λ f(x)+(1-x) f'(x)+x f''(x)=0

Solution

V4.0

DSolve[x f''[x]+(1-x) f'[x]+λ f[x]==0,f[x],x]

FunctionExpand[%]

V4.1

DSolve[x f''[x]+(1-x) f'[x]+λ f[x]==0,f[x],x]

V4.2

DSolve[x f''[x]+(1-x) f'[x]+λ f[x]==0,f[x],x]

V5.0

DSolve[x f''[x]+(1-x) f'[x]+λ f[x]==0,f[x],x]

V7

DSolve[x f''[x]+(1-x) f'[x]+λ f[x]==0,f[x],x]

Associative

Equation

λ f(x)+(ν+1-x) f'(x)+x f''(x)=0

Solution

V4.0

DSolve[x f''[x]+(ν+1-x) f'[x]+n f[x]==0,f[x],x]

V4.1

DSolve[x f''[x]+(ν+1-x) f'[x]+n f[x]==0,f[x],x]

V4.2

DSolve[x f''[x]+(ν+1-x) f'[x]+n f[x]==0,f[x],x]

V5.0

DSolve[x f''[x]+(ν+1-x) f'[x]+n f[x]==0,f[x],x]

V7

DSolve[x f''[x]+(ν+1-x) f'[x]+n f[x]==0,f[x],x]

Sum

1-Sum[((λ Pochhammer[1-λ,n-1])/(n!Pochhammer[ν+1,n]))(x^n),{n,3}]

1-Sum[((λ Pochhammer[1-λ,n-1])/(n!Pochhammer[ν+1,n]))(x^n),{n,∞}]

FullSimplify[Hypergeometric1F1[-λ,1+ν,x]-((λ! LaguerreL[λ,ν,x])/(Pochhammer[ν+1,λ]))]

FullSimplify[((λ! LaguerreL[λ,ν,x])/(Pochhammer[ν+1,λ]))]