Below I am given a coupled system of nonlinear ODEs:
I first considered equation $(1)$, and I defined some new functions.
$x_1 = h \implies x_1' = h' = x_2$
$x_2 = h' \implies x_2' = h'' = x_3$
$x_3 = h'' \implies x_3' = h''' = h''h + 0.5(h')^2 + 2g^2$
And likewise for equation (2), I defined some new function, $y_1$ to $y_2$, to get:
$y_2' = y_2h - y_1h'$
I was asked in the question to write $(1)$ and $(2)$ as a coupled system of nonlinear first-order ode's of the form:
$y' = f(y)$
I was hoping to know if I am on the right track so far? I am also thrown off by the part of the question where it asks me to write the coupled system in the form $y'=f(y)$.
Am I supposed to end up with one equation in the end, relation $h$ and $g$ ?
Thank you for your time