I am working on the following problem:

Let $x'=-ax+bf(y)$ and $y'=cx-df(y)$ with $f(0)=0$, $yf(y)>0$ for $y \neq 0$ and $a,b,c,d>0$. I need to show that the following function (for suitable values of p and q) is a strong Lyapunov function for the zero solutions of my system.

$$ V = px^2/2+q\int\_{0}^{y} f(u) \,du $$

I had no problem showing that $V>0$ for all nonzero input as it's easy to show that the integral is positive using the fact that $yf(y)>0$ and $f(0) = 0$

I then go on to try to show that the derivative of $V$ with respect to time is negative definite, which is where I run into problems.

$$\dot V = px\dot x + q\dot y f(y)$$

via the chain rule, but when I plug in the defined values of $\dot x$ and $\dot y$ I cannot figure out how to show that this expression is always less than zero. Any advice as to how to start this process?