Question

If a curve $y=f\left(x\right)$ passes through point $(1,-1)$ and satisfy the differential equation $y(1+xy)dx=xdy,$ then $f(-\frac{1}{2})$ equals

• A. $\frac{5}{4}$
• B. $\frac{4}{5}$
• C. $-\frac{4}{5}$
• D. $-\frac{5}{4}$

Answer 1

The correct option is B $\frac{4}{5}$

$y(1+xy)dx=xdy$
$\Rightarrow xdx+\frac{ydx-xdy}{y^2}=0$
$\Rightarrow d\left(\frac{x^2}{2}\right)+d\left(\frac{x}{y}\right)=0$
Integrating both sides, we get
$\frac{x^2}{2}+\frac{x}{y}=c$
It passes through point $(1,-1)$
$\Rightarrow c=-\frac{1}{2}$
$\Rightarrow \frac{x^2}{2}+\frac{x}{y}=-\frac{1}{2}$
$\therefore f(-\frac{1}{2})=\frac{4}{5}$