Question

Find the integrating factor and solve the following equations:

( 𝑦 − 𝑥^2) 𝑑𝑥 +( 𝑥^2sin 𝑦 − 𝑥 )𝑑𝑦 = 0

Answer 1

Try $\frac{1}{x^2}$ as an integrating factor:

$\left( \frac{y}{x^2}-1 \right) dx+\left( \sin y-\frac{1}{x} \right) dy=0\\d\left( -\frac{y}{x}-x-\cos y \right) =0\\\frac{y}{x}+x+\cos y=Const$