I found the differential equation from a textbook (which is about differential games):
$$ \phi'(x)= \frac{1-r-2x}{2(x-x^2)/\phi(x)-M-2} $$ with initial condition $\phi(0)=0$. The textbook just says that the solution is $$ \phi(x)=\frac{(1+r)x}{M+2} $$ without any derivations.
It is easy to verify that the $\phi(x)$ given above is just the solution. However, I have no idea how to obtain the solution from the differential equation. The standard techniques like separable methods, first order linear equations or method of characteristic seems not working here.