Solve the given initial-value problem. (x + y)2 dx + (2xy + x2 − 2) dy = 0, y(1) = 1

Let's check if the ODE is exact. To do that, we want to show that if



then . We have




so the equation is indeed exact. We're looking for a solution of the form . Computing the total differential yields the original ODE,




Integrate both sides of the first PDE with respect to ; then



where is a function of alone. Differentiate this with respect to so that





So the solution to this ODE is



i.e.

Horizontal change or the change of x