Question

(x² + y² + x) dx +xy dy = 0

Answer 1

For the equation (x^2 + y^2 + x)dx + xydy =0 , take xy=V(x) . Obtain

( x^2 + V^2/x^2 + x )dx + V( xdV - Vdx)/x^2 =0 . Then

(x^2 + x)dx + VdV/x =0.Separating

(x^3 + x^2)dx = -VdV . Integrating obtain the solution

6x^2y^2 + 4x^3 + 3x^4 = C