Find the first moment of a probability distribution governed by a nonlinear first order ODE

Question

May I ask if there is any standard way to find the first moment of a probability distribution governed by a nonlinear first-order ODE. For example, $$ \frac{\mathrm d p(x)}{\mathrm dx} = \alpha(x) p(x) $$ where $\alpha(x)$ is a non-linear function in terms of $x$. As far as I know, the most straightforward method is trying to solve the distribution $p(x)$ directly, and then find the first moment by $$ \langle x\rangle = \int x p(x)\mathrm dx $$ but sometimes the analytical form of this distribution is very hard to obtain, so instead of doing this is there other method?