# Solve a differential equation

1. Dec 18, 2009

### fraggle

1. The problem statement, all variables and given/known data

Solve this differential equation

http://www.texify.com/img/%5CLARGE%5C%21%5Cfrac%7B2%7D%7Bx%5E2%7D%28%5Cfrac%7Bdx%7D%7Bds%7D%29%5E2-%5Cfrac%7B1%7D%7Bx%7D%20%5Cfrac%7B%7Bd%5E2%7Dx%7D%7Bd%7Bs%5E2%7D%7D%3D1.gif [Broken]

(thanks Mark)

where x is unknown, s represents the metric http://www.texify.com/img/%5CLARGE%5C%21ds%5E2%3D%281/x%5E2%29dx%5E2%2B%281/x%5E2%29dy%5E2.gif [Broken]

2. Relevant equations
just that equation

3. The attempt at a solution

I tried writing out (d^2/ds^2)(1/x), but it wasn't right. x is unknown, s represents the metric ds^2=(1/x^2)dx^2+(1/x^2)dy^2

This might work if we use a substitution, I just don't know what.
Thanks

Last edited by a moderator: May 4, 2017
2. Dec 18, 2009

### Staff: Mentor

Last edited by a moderator: May 4, 2017
3. Dec 19, 2009

### fraggle

Figured it out.

Use
http://www.texify.com/img/%5CLARGE%5C%21%5Cfrac%7B%7Bd%5E2%7Du%28x%29%7D%7Bd%7Bs%5E2%7D%7D%3D%5Cfrac%7Bd%7D%7Bds%7D%20%28%5Cfrac%7Bdu%7D%7Bdx%7D%20%5Cfrac%7Bdx%7D%7Bds%7D%29.gif [Broken]
Oops, I guess I don't know how to paste in the text.

write out: d^2(u(x))/ds^2
then if we let u=(1/x) then plugging in we find that (d^2(1/x)/ds^2)=1/x

that's all I needed (solving for x)

Last edited by a moderator: May 4, 2017