I wonder if anyone has analytical insights into the following nonlinear ODE
$$(y^\prime)^2-{3\over2}y^2=-t^2$$
which occurs in cosmology (so called Hamilton-Jacobi equation).
I tried solving this numerically with RK4 (e.g. with some random initial condition, say $f(1)=1$). It is numerically sensitive, stemming from having to take the square-root (one can assume, say, that $y>0$ and $y^\prime>0$).
Does anyone have any analytical insights into simplifying this ODE, e.g., some substitution?
I'd very much appreciate your input. Thank you.
As the equation is a hyperbola equation, you can parametrize it as $$ y'(t)=t\sinh(u(t)), ~~~ y(t)=\sqrt{\frac23}t\cosh(u(t)). $$ Then the dynamic of $u$ is obtained by comparing the first equation to the derivative of the second equation $$ t\sinh(u(t)) = \sqrt{\frac23}(\cosh(u(t)) + t\sinh(u(t))u'(t)) \\~\\ \implies u'(t) = \sqrt{\frac32}-\frac{\cosh(u(t))}{t\sinh(u(t))} $$ which again gives singularities at $u(t)=0$, but perhaps these can be easier managed.