Question

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

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

Answer 1

The correct option is B $x^{3}y=2$

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