How to Solve a Differential Equation with Unknown x and Metric s

[fraggle]

Homework Statement

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


(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

Homework Equations

just that equation

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

 

[Mark44]

Added tex tags.

Homework Statement

Solve this differential equation (easiest to read if you plug the following equation into the link below):

$$\frac{2}{x^2}(\frac{dx}{ds})^2-\frac{1}{x} \frac{{d^2}x}{d{s^2}}=1$$

http://www.texify.com/links.php

or

(2/x^2)(dx/ds)^2 -(1/x)(d^2 x/ds^2)
where x is unknown, s represents the metric ds^2=(1/x^2)dx^2+(1/x^2)dy^2

Homework Equations

just that equation

The Attempt at a Solution

I tried writing out (d^2/ds^2)(1/x), but it wasn't right.

Homework Statement

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

 

[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
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)