Question

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

• A. $\frac{x}{y}+e^{x^3}=c$
• B. $\frac{x}{y}-e^{x^3}=c$
• C. $\frac{x}{y}=-e^{x^3}+c$
• D. none of these

Answer 1

The correct option is A $\frac{x}{y}+e^{x^3}=c$

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

Consider $\int \frac{ydx-xdy}{y^2}$

We know that $\frac{d}{dx}\left(\frac{x}{y}\right)=\frac{ydx-xdy}{y^2}$

$\int \frac{d}{dx}\left(\frac{x}{y}\right)dx$

$=\frac{x}{y}+c_1$

Consider $3\int x^2{e}^{x^3}dx$

$=\int 3x^2{e}^{x^3}dx$

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

$=\int e^{t}dt$

$=e^t+c_2$

$=e^{x^3}+c_2$

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

$\frac{x}{y}+c_1+e^{x^3}+c_2=0$

$\Rightarrow \frac{x}{y}+e^{x^3}=-(c_1+c_2)$

$\Rightarrow \frac{x}{y}+e^{x^3}=c$ where $c=-(c_1+c_2)$