solve the given DE

Question

xy*y' =x²+y²

*its a homogenus DE with degree zero

Answer 1

This is homogenius equation. Sabstitution: y = xv; y' = v + xv'

x²v(v + xv') = x² + x²v²; After reducing by x² we get v² + xvv' = 1 + v²; xvv' = 1;vdv = dx/x; ∫vdv = dx/x

v²/2 = lnIxI + C/2; v² = 2lnIxI + C; y² = x²(2lnIxI + C)∫