Non Linear ODE and change of variable

Jun 2009
696
170
United States
I am trying to write the nonlinear ODE below as a first order ODE using the coordinate transformation (x,y) --> (t,u)...

\(\displaystyle y''=x^{-\frac{1}{2}}y^{\frac{3}{2}}\) using \(\displaystyle t=y(x)^{\frac{1}{6}}\frac{\sqrt{x}}{144^{\frac{1}{6}}}\) and \(\displaystyle u(t)=-\left(\frac{16}{3}\right)^{\frac{1}{3}}\frac{y'(x)}{y(x)^{\frac{3}{4}}}\)

This is how the ODE and change of variables are given exactly in the problem. The notation is also exact. I asked my instructor about the u(t) function since it doesn't make sense to me but he didn't seem to understand and kept saying it was a coordinate transformation. My objection was that the notation implies u(t) is a function of t only. But maybe it's just meant to be u instead. but those are the variable changes I'm instructed to use. I have no idea how to proceed and my attempts so far went nowhere.

My instinct was to differentiate the u equation since that could result in the first order derivative for u in terms of a second order derivative for y (among other variables) but I don't know how to apply the chain rule to u(t). Is it u(t) a composite like u=y(x(t))?
Any nudge would be appreciated.