Try letting . Then the DE becomes
This is now a Bernoulli DE which should easily be solvable
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?