Solve the given initial-value problem. (y2 cos x − 3x2y − 4x) dx + (2y sin x − x3 + ln y) dy = 0, y(0) = e

The ODE is exact if . We have




and so the equation is indeed exact. So we're looking for a solution of the form , noting that the total differential for this solution is



which corresponds to



In the first PDE, we can integrate both sides with respect to to get




where is assumed to be a function of alone. Then differentiating with respect to gives us





So the general solution is



or simply



Given that , we have




so the particular solution to the IVP is