Question

Non-Exact Differential Equations. Find the complete solution of ydx + (x + y ^ 2) dy = 0

Answer 1

$$ydx+(x+y^2)dy=0$$

$$\dfrac{dx}{dy}+\dfrac{1}{y}x=-y$$

$$I.F.=e^{\int(1/y)dy}=y$$

$$ydx+xdy=-y^2dy$$

$$d(xy)=-y^2dy$$

Integrate

$$\int d(xy)=-\int y^2dy$$

$$xy=-\dfrac{y^3}{3}+C$$

$$xy+\dfrac{y^3}{3}=C$$