Solve the Differential equation $$\frac{dy}{dx}=\frac{5x^3-xy^2-2x}{3x^2y-y^3}$$
My try: Let $x^2=X$ and $y^2=Y$ we get
$xdx=dX$ and $ydy=dY$ then
$$\frac{dY}{dX}=\frac{5X-Y-2}{3X-Y}$$ which is a Non homogenous Differential equation.Is there any formal approach to solve this?
Make the $(u,v)$ substittion. Such that you have at the numerator $au+bv$ and at the denominator $cu+dv$ , where $a,b,c,d$ are constants. We want to have this form: $$\frac {dv}{du}=\frac {au+bv}{cu+dv}$$
For this purpose , we try this substitution $$X=u+a$$ $$Y=v+b$$ After basic calculation, we get $(a,b)=(1,3)$ and therefore : $$X=u+1$$ $$Y=v+3$$
The equation becomes $$\frac {dv}{du}=\frac {5u-v}{3u-v}$$
This equation is homogeneous. Susbtitute $$v=tu \implies v'=t'u+t $$