Simple ODE question

JuanFangio · October 20th, 2008, 1:06 pm

Hi gals and guys,I was wondering if someone could help me with the following ODE question:Show that if n belongs to Z the only solutios of the differential equation(r^2)F''(r) + rF'(r) - (n^2)F(r) =0 which are twice differentiable when r>0, are given by linear combinations of r^n and r^-n and 1 and log (r) when n=0Thanks in advance!

ppauper · October 20th, 2008, 1:21 pm

it's a second order linear ODE so you can show that it has exactly 2 linearly independent solutions (existence and uniqueness theorems)Then show that the solutions you gave are solutions, and you're done.

JuanFangio · October 20th, 2008, 5:32 pm

Thanks ppauper. Here's my attempt with n=0(r^2)y'' + ry' = 0s^2 + s = 0 , so s=0. s= -1y= c1*exp(-r) + c2*exp(0) = c1exp(-r) + c2I'm trying to use the standard formula for homogeneous, second order PDEs. Is this an incorrect approach?Thanks again!

ppauper · October 20th, 2008, 6:03 pm

QuoteOriginally posted by: JuanFangioThanks ppauper. Here's my attempt with n=0(r^2)y'' + ry' = 0s^2 + s = 0 , so s=0. s= -1y= c1*exp(-r) + c2*exp(0) = c1exp(-r) + c2I'm trying to use the standard formula for homogeneous, second order PDEs. Is this an incorrect approach?either i) seek solutions y=r^korii) make the transformation r=e^x, x=log(r) and you have a constant coefficient equation