Question

Find the equation of the curve passing through the point $(0,\frac{\pi }{3})$ and satisfying the differential equation $\sin x\cos ydx+\cos x\sin ydy=0,$ where$x,y\in (0,\frac{\pi }{2})$

• A. $\sec \frac{x}{y}=2$
• B. $\sec xy=2$
• C. $\sec x\cdot \sec y=2$
• D. $\sec \frac{y}{x}=2$

Answer 1

The correct option is C $\sec x\cdot \sec y=2$

Given differential equation
$\sin x\cos ydx+\cos x\sin ydy=0$
$\Rightarrow -\tan xdx=\tan ydy$
Integrating both sides, we get
$-\log \sec x+\log C=\log \sec y$
$\Rightarrow \sec x\cdot \sec y=C\cdots \left(1\right)$
Equation $\left(1\right)$ passes through $(0,\frac{\pi }{3}).$
$\Rightarrow C=2$
Therefore, equation of the curve is $\sec x\cdot \sec y=2$