Can't solve differential equation

Question

I tried making this substitution using what I know about the homogeneity of this ODE:

Friend,where am I going wrong here?

Answer 1

Hint:

The sustitution $x=vy,\;dx=vdy+ydv\,$ give us \begin{align*} &&-y(vdy+ydv)+\left(vy+\sqrt{vy^2}\right)dy&=0\\ \iff&& -y^2dv+\sqrt{vy^2}dy&=0 \end{align*} Now, let's suppose $y> 0$ and $v>0$, then we get $$-y^2dv+y\sqrt v\, dy=0$$ And then $$\dfrac{dy}{y}=\dfrac{dv}{v^{1/2}}$$

Answer 2

Integrating factor is $$\mu=\frac{1}{xP+yQ}=\frac{1}{x(-y)+y(\sqrt{xy}+x)}=\frac{1}{y\sqrt{xy}}$$ Then solution of differential equation $-ydx+(x+\sqrt{xy})dy=0$ is $$\log{(y)}-2 \sqrt{\frac{x}{y}}=C$$