Use separation of variables to find the solution to the differential equation $\frac{dz}{dy}=zy$ if $z=4$ when $y=0$.
$$z(y)=$$

Step 1 ：The given differential equation is a first order linear differential equation. The method of separation of variables involves rearranging the equation so that all terms involving $z$ are on one side and all terms involving $y$ are on the other side. Then, we can integrate both sides with respect to their respective variables.

Step 2 ：The solution to the differential equation is given by the function $f(y)=C{e}^{\frac{y^2}{2}}$, where $C$ is a constant. We can find the value of $C$ by substituting the initial condition $z=4$ when $y=0$ into the solution.

Step 3 ：The solution to the equation $C{e}^0=4$ is $C=4$. Therefore, the solution to the differential equation with the given initial condition is $z(y)=4e^{\frac{y^2}{2}}$.

Step 4 ：Final Answer: $z(y)=\boxed{{\displaystyle 4e^{\frac{y^2}{2}}}}$