Second-order ODE, reduction of order?

Find the specified particular solution:

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The equation seems amenable to the substitution , so it can be transformed into , or . Since , the latter equation is exact, and is solvable through the method of exact equations. Unfortunately, this yields the rather cumbersome expression for some constant , which I would have a hard time integrating, I think.

Any clever tricks?

Re: Second-order ODE, reduction of order?

Try letting . Then the DE becomes

This is now a Bernoulli DE which should easily be solvable :)

Re: Second-order ODE, reduction of order?

Nice approach! I'll give it a go in a little while. Also, I think I was just being retarded when I posted this (and possibly still am), as I realized that , and so , and thus, . Ugh.