Answer: Discuss how the functions M(x, y) and N(x, y) can be found so that each

Chapter 2, Problem 43

(choose chapter or problem)

Differential equations are sometimes solved by having a clever idea. Here is a little exercise in cleverness: Although the differential equation

\(\left(x-\sqrt{x^{2}+y^{2}}\right) d x+y d y=0\)

is not exact, show how the rearrangement

\(\frac{x d x+y d y}{\sqrt{x^{2}+y^{2}}}=d x\)

and the observation \(\frac{1}{2} d\left(x^{2}+y^{2}\right)=x d x+y d y\) can lead to a solution.