Lommel Differential Equation

Author

Eric W. Weisstein
February 17, 2008

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

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

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

Lommel Functions

LommelS1[μ_,ν_,z_]:=(z^(μ+1)/((μ+1)^2-ν^2))*HypergeometricPFQ[{1},{(3+μ+ν)/2,(3+μ-ν)/2},-(z^2/4)]

LommelS2[μ_,ν_,z_]:=(z^(μ+1)/((μ+1)^2-ν^2))*HypergeometricPFQ[{1},{(3+μ+ν)/2,(3+μ-ν)/2},-(z^2/4)]+((2^(-1+μ+ν)*Gamma[ν]*Gamma[(1/2)*(1+μ+ν)])/(z^ν*Gamma[(1/2)*(1-μ+ν)]))*Hypergeometric0F1[1-ν,-(z^2/4)]+((2^(-1+μ-ν)*z^ν*Gamma[-ν]*Gamma[(1/2)*(1+μ-ν)])/Gamma[(1/2)*(1-μ-ν)])*Hypergeometric0F1[1+ν,-(z^2/4)]

LommelS1[μ,ν,z]//TraditionalForm

LommelS2[μ,ν,z]//TraditionalForm

LommelS2[μ,ν,z]//FullSimplify//TraditionalFOrm

TraditionalForm[%]

FullSimplify[(((2^(μ+ν-1)) Γ(ν) Γ((1/2) (μ+ν+1)) (Subscript[, 0])(Subscript[F, 1])(\[Null];1-ν;-((z^2)/4)) (z^(-ν)))/(Γ((1/2) (-μ+ν+1))))+(((2^(μ-ν-1)) Γ((1/2) (μ-ν+1)) Γ(-ν) (Subscript[, 0])(Subscript[F, 1])(\[Null];ν+1;-((z^2)/4)) (z^ν))/(Γ((1/2) (-μ-ν+1))))]//TraditionalForm

ODE

(((∂^2)y1(μ,ν,z))/(∂(z^2))) (z^2)+((∂y1(μ,ν,z))/(∂z)) z+((z^2)-(ν^2)) y1(μ,ν,z)==(z^(μ+1))

y1[μ_,ν_,z_]:=C1*BesselJ[ν,z]+C2*BesselY[ν,z]+LommelS1[μ,ν,z]

Chop[Collect[z^2*D[y1[μ,ν,z],{z,2}]+z*D[y1[μ,ν,z],{z,1}]+(z^2-ν^2)*y1[μ,ν,z]-z^(μ+1)/.{ν->Random[Complex]}/.{μ->Random[Complex]}/.{z->Random[Complex]},C1]]

(((∂^2)y2(μ,ν,z))/(∂(z^2))) (z^2)+((∂y2(μ,ν,z))/(∂z)) z+((z^2)-(ν^2)) y2(μ,ν,z)==(z^(μ+1))

y2[μ_,ν_,z_]:=C1*BesselJ[ν,z]+C2*BesselY[ν,z]+LommelS2[μ,ν,z]

Chop[Collect[z^2*D[y2[μ,ν,z],{z,2}]+z*D[y2[μ,ν,z],{z,1}]+(z^2-ν^2)*y2[μ,ν,z]-z^(μ+1)/.{ν->Random[Complex]}/.{μ->Random[Complex]}/.{z->Random[Complex]},C1]]

DSolve

(DSolve[(z^2)y''[z]+z y'[z]+((z^2)-(ν^2))y[z]==k (z^(μ+1)),y[z],z]//FullSimplify)//TraditionalForm

(DSolve[(z^2)y''[z]+z y'[z]+((z^2)-(ν^2))y[z]== (z^(μ+1)),y[z],z]//FullSimplify)//TraditionalForm

(DSolve[(z^2)y''[z]+z y'[z]+((z^2)-(ν^2))y[z]==- (z^(μ+1)),y[z],z]//FullSimplify)//TraditionalForm

Also Known As Lommel

DSolve[y''[z]+g[y[z]]((y'[z])^2)+f[x]y'[z]==0,y[z],z]