Question

Solve $(x^2-y^2)dx-xydy=0$

Answer 1

$(x^2-y^2)dx-xydy=0$

$\frac{dy}{dx}=\frac{x^2-y^2}{xy}$

Let $y=vx$, $\frac{dy}{dx}=v+x\frac{dv}{dx}$

$v+x\frac{dv}{dx}=\frac{x^2-v^2{x}^2}{v{x}^2}=\frac{1-v^2}{v}$

$x\frac{dv}{dx}=\frac{1-v^2-v^2}{v}$

$\int \frac{v}{1-2v^2}dv=\int \frac{dx}{x}$

Let $v^2=t2vdv=dt$

$\int \frac{dt}{2(1-2t)}=\int \frac{dx}{x}$

$\frac{-1}{4}ln(1-2t)=lnx+c$

$4lnx+ln(1-2t)+c=0$

$ln{x}^4+ln(1-2v^2)+c=0$

$ln{x}^4+ln(1-2\frac{y^2}{x^2})+c=0$

$ln{x}^4+ln(x^2-2y^2)-ln(x{)}^2+c=0$

$ln{x}^2+ln(x^2-2y^2)+c=0$

$2lnx+ln(x^2-2y^2)+c=0$