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