If I have the implicit general solution $y^{3/2} = \dfrac{-3}{20} [ \cos(x) + 3\sin(x)] + Ce^{3x}$, then am I correct in saying that the explicit general solution is $y = \left\{ \dfrac{-3}{20} [ \cos(x) + 3\sin(x)] + Ce^{3x} \right\}^{2/3}$?
I just want to make sure, since, for some reason, my solutions only list the implicit general solution and not the explicit general solution.
$y^{3/2}$ is not implicit, we use the term implicit when it is impossible or extremely hard or at least hard to give a function its clear expression with respect to $x$, in your case the expression is already found. Just a simple taking to the power of $2/3$ so you are right, nothing wrong with your passage.