In Sean Carroll's Introduction to General Relativity: Spacetime and Geometry, the section under Birkhoff's theorem in Chapter 5 attempts to remove cross terms of the form $(dtdr + drdt)$ in the metric by redefining coordinates. The original metric reads $$ds^2 = g_{aa}(a, r)da^2 + g_{ar}(a,r)(dadr + drda) + g_{rr}(a,r)dr^2 + r^2d\Omega^2.$$ He does so by defining a coordinate $t(a, r)$ with $dt = \partial_at da + \partial_rtdr$, so that $dt^2 = (\partial_at)^2da^2 + \partial_at\partial_rt (dadr+drda) + (\partial_rt)^2dr^2$. Insisting that this make the metric become $$ds^2 = m(t,r)dt^2 + n(t,r)dr^2$$ gives us three equations for three unknowns $m(a,r)$, $n(a,r)$ and $t(a,r)$: $$m(\partial_at)^2 = g_{aa},$$ $$n + m(\partial_rt)^2 = g_{rr}$$ and $$m(\partial_at)(\partial_rt) = g_{ra}.$$
Now, I can see quite easily how we can solve for $n$, given that we have the right hand sides. This is obtained by subtracting $n$ in equation 2, then multiplying equations 1 and 2, and dividing by the square of equation 3. Upon solving for $n$ explicitly, the only non-trivial equations are $$m(\partial_at)^2 = g_{aa}$$ and $$m(\partial_rt)^2 = g_{rr} - n.$$
Given initial conditions for $t$, how would one go about solving them? I do not quite know how to deal with simultaneous equations involving the partial derivatives of a quantity.
