Chapter 2.4, Problem 43E

Differential equations are sometimes solved by having a clever idea. Here is a little exercise in cleverness: Although the differential equation $\left(x-\sqrt{x^2+y^2}\right)dx+ydy=0$ is not exact, show how the rearrangement (x dx + y dy)/ $\sqrt{x^2+y^2}$ = dx and the observation $\frac{1}{2}$d(x² + y²) = x dx + y dy can lead to a solution.