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

Homework Equations

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,

 

[dextercioby]

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.

 

[dextercioby]

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 a lot 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?

 

[dextercioby]

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.