This is actually a very easy problem but I am brand new to this subject and I just don't the mechanism on how to do it yet. The question is:
Determine the solution of $\frac{\partial \rho}{\partial t}=(\sin x)\rho$ which satisfies $\rho (x,0)=\cos x$.
First, forget the fact that $\rho$ is a function of two variables and just look at it as a function of $t$. If I tell you that $\sin x$ is a constant $A$ and $\cos x$ just some constant $B$, how would you solve the ODE $$\frac{d\rho}{dt}=A\rho\\\rho(0) = B,$$
where $\rho$ is just a function of time?
Once you do that, plug $A$ and $B$ for what they are and you will have $\rho(x,t).$