I'm trying to figure out a way to solve the following second-order nonlinear ODE:
with the initial conditions
I've considered multiplying both sides by $y'$, but that doesn't seem particularly illuminating (for me at least) due to the $y'$ and the $t$ already in the problem.
I've also thought about maybe some kind of integrating factor, or maybe making a substitution like $u=y^2$ to make this linear, but that only seems to complicate things. Currently, I'm thinking that I might need to convert this to a first-order vector equation, but even doing so has me a little stumped when dealing with the resulting first-order scalar equations.
Any thoughts? I also welcome any general recommendations for resources on solving second-order differential equations. I'm good with second-order linear homogeneous equations with constant coefficients, but once it gets more complicated than that, I panic. Thank you!