coordinate transformation to flat space

stampita

1. The problem statement, all variables and given/known data
find the transformation that turns this metric:
[tex]ds^2=-X^2dT^2+dX^2[/tex]

into the minkowski metric: diag(-1,1).

3. The attempt at a solution
attempt 1:
I transformed the above metric into the coordinates that use the lines that define the light cone as the axes. Namely,
[tex]u=T+ln(X)[/tex]
[tex]v=T-ln(X)[/tex]

attempt 2:
If I can transform the above, into the coordinates u and v such that i have
[tex] ds^2=-dudv [/tex]
then I'm done because with another transformation:
[tex]dt'=\frac{du+dv}{2}[/tex]
[tex]dx'=\frac{du-dv}{2}[/tex]
I have the relation
[tex]-(dt')^2+dx^2=-(\frac{du+dv}{2})^2+(\frac{du-dv}{2})^2=-dudv.[/tex]

Is there a systematic way of doing this?