Question

If the curve $y=f(x)$ passing through the point $(1,2)$ and satisfies the differential equation $xdy+\left(y+x^3{y}^2\right)$ $dx=0$, then

• A. $xy=\frac{1}{2}$
• B. $x^{3}y=2$
• C. $\frac{1}{xy}=2$
• D. None of these

Answer 1

Answer: (B)

$xdy+\left(y+x^3{y}^2\right)dx=0$
$\Rightarrow xdy+ydx=-x^3{y}^{2}dx$
$\Rightarrow \frac{xdy+ydx}{x^2{y}^2}=-xdx$
$\Rightarrow \frac{d(xy)}{(xy{)}^2}=-xdx$
Integrating, we get
$-\frac{1}{xy}=-\frac{x^2}{2}+c...$(1)
Using $(1,2)$ in $(1)$, we get
$-\frac{1}{2}=-\frac{1}{2}+c\Rightarrow c=0$
$\therefore -\frac{1}{xy}=-\frac{x^2}{2}$
$\Rightarrow y=\frac{2}{x^3}$ or $x^{3}y=2$
is the required curve.