Questionable solution from DSolve
- To: mathgroup at smc.vnet.net
- Subject: [mg63629] Questionable solution from DSolve
- From: dkjk at bigpond.net.au
- Date: Sun, 8 Jan 2006 03:32:49 -0500 (EST)
- Sender: owner-wri-mathgroup at wolfram.com
Hi group,
Consider the differential equation
D[f[r,z], {r, 2}] + (1/r)D[f[r, z], {r, 1}] - (kz^2 + 1^2)f[r, z] ==
g[r,z]
With the RHS set to 0, the solution is simply
BesselJ[0, I*r*Sqrt[kz^2+1]]C[1][z] + BesselY[0,
-I*r*Sqrt[kz^2+1]]C[2][z]
as expected. But if I put g[r,z] = - Exp[-a * r^2 - b * z^2]
I obtain
BesselJ[0, I*r*Sqrt[kz^2+1]]C[1][z] + BesselY[0,
-I*r*Sqrt[kz^2+1]]C[2][z]
plus some integral with 1 in the lower terminal and r in the upper
terminal. This is very odd, since the solution is not even
dimensionally correct! The variable r can be considered to have units
of metres whereas unity is dimensionless. You might claim that the
original differential equation is not dimensionally corect to begin
with, but multiply any of the terms by arbitrary constants and you
still end up with unity in the lower terminal.
Is this solution rubbish, or have I overlooked something ?
Thanks in advance.
James
DSolve[D[f[r,
z], {r, 2}] + (1/r)D[f[r, z], {r, 1}] - (
kz^2 + 1^2)f[r, z] == -Exp[-a*r^2 - b*z^2], f[r, z], {r, z}]