Factorising and integrating a differential

[eddysd]

Having a bit of trouble with this equation, I need to find V explicitly and this would obviously be done by factorising and integrating, but I can't seem to factorise it correctly. I have what I think is the correct answer but can't do the steps to get there. Any help would be greatly appreciated.

δ²V/δr² + 1/r δV/δr - V/r = 0

which I think goes to:

V = C₁r/2 + C₂/r

C₁ and C₂ being constants of integration.

 

[tiny-tim]

hi eddysd!

δ²V/δr² + 1/r δV/δr - V/r = 0

which I think goes to:

V = C₁r/2 + C₂/r

no, i think that would be δ²V/δr² + 1/r δV/δr - V/r² = 0

 

[eddysd]

hi eddysd!


no, i think that would be δ²V/δr² + 1/r δV/δr - V/r² = 0

Yes sorry, typo, that is the correct equation! Do you think you could tell me how to get from that to the equation for V?

 

[JJacquelin]

δ²V/δr² + 1/r δV/δr - V/r = 0

Solutions involves Bessel functions.

 

[andrien]

substiution r=exp(y) will give you the required answer .To know why multiply the eqn by r^2 and see the second term and it does not involve bessel functions.

 

[JJacquelin]

Yes sorry, typo, that is the correct equation! Do you think you could tell me how to get from that to the equation for V?

Now it is an homogeneous ODE.
So, use the classical method : Let V=r*W and rewite the ODE where W is the unknown function.