The general solution of the differential equation  $(x-y^2)dx+y(5x+y^2)dy=0$  is 

$$\begin{gathered} y\frac{dy}{dx}=\frac{y^2-x}{5x+y^2} \\ let y^2=vx \\ 2y\frac{dy}{dx}=v+x\frac{dv}{dx} \\ \frac{1}{2}(v+x\frac{dv}{dx})=\frac{vx-x}{5x+vx} \\ v+x\frac{dv}{dx}=2\left(\frac{v-1}{5+v}\right) \\ x\frac{dv}{dx}=-\frac{v^2+3v+2}{5+v} \\ \int \frac{5+v}{(v+1)(v+2)}dv=-\int \frac{dx}{dx} \\ \int (\frac{4}{v+1}-\frac{3}{v+2})dx=-\int \frac{dx}{x} \\ 4\log |v+1|-3\log |v+2|=-\log x+\log c \end{gathered}$$