Use separation of variables to find the solution to the differential equation $\frac{d z}{d y}=z y$ 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 \(z(y) = 4 e^{\frac{y^2}{2}}\).

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