Solving differential equations with an integral

Hi people,

I have a problem that I'm not able to find a proper solution. Imagine a population balance, we have P1 particles of size v1, P2 particles of size v2, Pn particles of size vn. So P is a kind of histogram that is time dependent. v can be fixed in advance.

We have the following system of differential equations

dM/dt= M int(v*P)

dP/dt= M d(v*P)/dv

Where int(v*P) is the integral of v*P from 0 to infinity.

After solving the system M must be a vector and P must be a matrix, in which each row is the histogram at any time.

Could you please indicate me how to solve these kind of problems? Currently I am solving the first equation in a for loop and the second one inside the loop. However, this is very time consuming and I am not sure if the solution is correct.

Thank you very much in advance.