Question

Solve $x(1-x^2)dy+(2x^{2}y-y-5x^3)dx=0$.

Answer 1

Solve $x(1-x^2)dy+(2x^{2}y-y-5x^3)dx=0$

$x(1-x^2)dy=(5x^3-2x^{2}y+y)dx$

$\frac{dy}{dx}=\frac{5x^3+y-2x^{2}y}{x(1-x^2)}$

$\frac{dy}{dx}=v+x\frac{dv}{dx}$

$v+x\frac{dv}{dx}=\frac{5x^3+vx-2x^2\left(vx\right)}{x(1-x^2)}$

$x\frac{dv}{dx}=\frac{5x^2+v-2x^{2}v}{1-x^2}-v$

$x\frac{dv}{dx}=\frac{5x^2+v-2x^{2}v-v+v{x}^2}{1-x^2}$

$\frac{dv}{dx}=\frac{(5-v)x^2}{x(1-x^2)}$

$\frac{dv}{5-v}=\frac{x}{1-x^2}dx$

$\frac{dv}{v-5}=\frac{x}{x^2-1}dx$

Put $x^2-1=t=>2xdx=dt$

and integral both side

$\ln |v-5|=\frac{1}{2}\int \frac{dt}{t}=\frac{1}{2}\ln \left|t\right|+\ln c$

$\ln |v-5|=\ln \sqrt{t}+\ln c$

$\ln \left|\frac{y-5x}{x}\right|=\ln c\sqrt{1-x^2}$

$y=5x+cx\sqrt{1-x^2}$