I am trying to solve this differential equation which popped up in an engineering problem.
\begin{align} &a\dot{V}(t) + b P(t) &= x(t)\\ &V(t)P(t) &= y(t) \end{align}
The values $a$ and $b$ are known constants and $x(t)$ and $y(t)$ are known functions. We are asked to find $V(t)$ and $P(t)$. The equation was more complicated in the beginning and this is as much as I managed to simplify it, however I am stuck at this point, and I haven't solved differential equations in a while.
It's not hard to reduce this to $$ \frac{x(t)}{b}V(t) - \frac{a}{b}V(t)\dot{V}(t) = y(t) $$ But this is non-linear and I don't know how to solve it.
Your ODE is an Abel equation of the second kind. Here's a resource for finding your solution! http://eqworld.ipmnet.ru/en/solutions/ode/ode0125.pdf