Helmholtz Differential Equation—Polar Coordinates

Author

Eric W. Weisstein
May 3, 2004

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

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

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

Solution

Angular

DSolve[(1/(Θ[θ]))Θ''[θ]==-(m^2),Θ[θ],θ]

Radial

DSolve[((r^2)/(R[r]))R''[r]+(r/(R[r]))R'[r]+(k^2)(r^2)==m^2,R[r],r]

Check

With[{f=(C[1] Cos[m θ]+C[2] Sin[m θ])(BesselJ[m,k r] C[3]+BesselY[m,k r] C[4])},
(1/r)(∂_r)(r (∂_r)f)+(1/(r^2))(∂_(θ,θ))f+(k^2)f
]//FullSimplify