Show that (2xy3 + cos x)dx + (3x2 y2 − sin y)dy = 0 is exact, and find the solution. find c if y(0) = π

We're looking for a solution of the form . By the chain rule, this solution should have total differential

and the equation is exact if the mixed second-order partial derivatives of are equal, i.e. .

The given ODE is exact, since

Then

With , we get

by definition, an arc subtended by a central angle of some given measure has the same measure. so .