Question

The solution of y $dx-xdy+3x^2{y}^2{e}^{x^3}dx=0$ is

Answer 1

$ydx-xdy+3x^2{y}^2{e}^{x^3}dx=0$

$\Rightarrow ydx-xdy=-3x^2{y}^2{e}^{x^3}dx$

$\Rightarrow \frac{ydx-xdy}{y^2}=-3x^2{e}^{x^3}dx$

$d\left(\frac{x}{y}\right)=\frac{ydx-xdy}{y^2}$ (standard differentials)

$\Rightarrow d\left(\frac{x}{y}\right)=-3x^2{e}^{x^3}dx$

Integrating both sides.

$\Rightarrow \int d\left(\frac{x}{y}\right)=-\int 3x^2{e}^{x^3}dx$

Let $e^{x^3}=t$

$\Rightarrow \left(e^{x^3}\right)\left(3x^2\right)=\frac{dt}{dx}$

$\Rightarrow dt=3x^2{e}^{x^3}dx$

$\Rightarrow \int d\left(\frac{x}{y}\right)=-\int dt$

$\Rightarrow \frac{x}{y}=-t+c$

$\Rightarrow \frac{x}{y}+t=c$

$\Rightarrow \frac{x}{y}+e^{x^3}=c$