Determine if the differential equation $y^{\prime}=x e^{y}$ is separable, and if so, write it in the form $h(y) d y=g(x) d x$.
NOTE: If the equation is not separable, indicate with the checkbox.
$d y=$ $d x$ Not separable.
Answer
Final Answer: The differential equation \(y^{\prime}=x e^{y}\) is separable and can be written in the form \(\boxed{e^{-y} dy = x dx}\).
Step 1 ：The given differential equation is \(y^{\prime}=x e^{y}\). A differential equation is said to be separable if it can be written in the form \(h(y) dy = g(x) dx\).
Step 2 ：In this case, we can see that the function \(y^{\prime}\) is a product of a function of \(x\) and a function of \(y\). Therefore, it seems that the equation is separable.
Step 3 ：We can write it in the form \(h(y) dy = g(x) dx\) by dividing both sides by \(e^{y}\) and multiplying both sides by \(dx\). This gives us \(h(y) = e^{-y}\) and \(g(x) = x\).
Step 4 ：Final Answer: The differential equation \(y^{\prime}=x e^{y}\) is separable and can be written in the form \(\boxed{e^{-y} dy = x dx}\).
