# Inhomogeneous second order ODE with non-constant coefficient

jmz34

## Homework Statement

Solve ODE of form y''+(2/x)y'=C*(e^y) where C is a constant

## The Attempt at a Solution

I don't really see how to approach this one, so a point in the right direction would be great.

Thanks,

Homework Helper
Where did you get this problem from ? I'm almost sure there's no analytical solution (Mathematica couldn't find any combination of elementary/special functions which would satisfy the equation) to it, so an approximate solution would be the best you could ask for.

jmz34
Where did you get this problem from ? I'm almost sure there's no analytical solution (Mathematica couldn't find any combination of elementary/special functions which would satisfy the equation) to it, so an approximate solution would be the best you could ask for.

I was trying to solve del^2(Psi)=Ae^(Psi) in spherical polars, for the radial component.

Checking over my algebra I'm pretty sure it's correct.

Homework Helper
Yes, it looks correct. However, my guess is that the nonlinearity in the RHS spoils the <neat> intregrability.

jmz34
Yes, it looks correct. However, my guess is that the nonlinearity in the RHS spoils the <neat> intregrability.

Thanks alot for your help. The question does say that Psi varies over a length scale that is approximately the same as the region which I'm supposed to be analyzing. Does that somehow mean I can take the RHS as constant?

Homework Helper
Not really, rather approximating it to y(x). That way you get an ODE which can be integrated in terms of elementary functions.

And the C needs to be specified, the sign of it is important. You may rescale it to +1 or -1, I'm sure.

As for how to solve the following 2 ode's

$$y''+ \frac{2}{x}y'\pm y = 0$$

use the substitution

$$y(x) = \frac{u(x)}{x}$$

You'll find 2 classes of solutions, depending on the sign of the rescaled constant.

jmz34
Ofcourse. Thanks again.