Differential equations are sometimes solved by having a clever idea Here is a little exercise in cleverness Although the differential equation x (x 2 y 2 ) dx y dy 0 is not exact, show how the rearrangement (x dx y dy) (x 2 y 2 ) dx and the observation d(x 2 y 2 ) x dx y dy can lead to a solution ...

First note that Then x dx y dy x 2 y 2 dx becomes ...

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

Question:

Differential equations are sometimes solved by having a clever idea. Here is a little exercise in cleverness: Although the differential equation {x – √(x² + y²)} dx + y dy = 0 is not exact, show how the rearrangement (x dx + y dy)/√(x² + y²) = dx and the observation ½ d(x² + y²) = x dx + y dy can lead to a solution.