Anger Differential Equation

Author

Eric W. Weisstein
January 20, 2004

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

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

©2005 Wolfram Research, Inc. except for portions noted otherwise

Solution

V6.1

DSolve[y''[x]+((y'[x])/x)+(1-((ν^2)/(x^2)))y[x]==((x-ν)/(π (x^2)))Sin[π ν],y[x],x]

FullSimplify[%]

y''[x]+((y'[x])/x)+(1-((ν^2)/(x^2)))y[x]==((x-ν)/(π (x^2)))Sin[π ν]/.y:>Function[x,(2^(-2-ν)) (x^(-ν)) ((2^(2+ν)) (x^ν) (BesselJ[ν,x] C[1]+BesselY[ν,x] C[2])+(1/(Gamma[2-ν]))(4^ν) BesselJ[ν,x] (2 x HypergeometricPFQ[{(1/2)-(ν/2)},{1-ν,(3/2)-(ν/2)},-((x^2)/4)]-2 (-1+ν) HypergeometricPFQ[{-(ν/2)},{1-ν,1-(ν/2)},-((x^2)/4)])+(x^(2 ν)) BesselJ[-ν,x] (ν Gamma[(ν/2)] HypergeometricPFQRegularized[{(ν/2)},{1+ν,1+(ν/2)},-((x^2)/4)]-x Gamma[((1+ν)/2)] HypergeometricPFQRegularized[{((1+ν)/2)},{1+ν,((3+ν)/2)},-((x^2)/4)]))]//FullSimplify