Let $$\dot p = (1-\mu)$$ $$\dot \mu = (p - A - B\mu)\mu$$ with $A, B$ constants with $B\geq 0$ and $A + B \geq 0$. Assume $\mu >0$. Find a Lyapunov function L of the form $ L = \frac{p^2}{2} + C_1p + C_2(\mu - \ln(\mu))$.
The fixed point is $(\mu,p) = (1, A+B)$. $\dot L = C_2(\mu-1)(p-A-B\mu) + (p+C_1)(1-\mu)$. $\dot L(1, A+B) = 0$ as required. We will find a weak Lyapunov function. I tried a few things to find $C_1$ and $C_2$ but everything led to an dead end. Can someone give a hint?