Question
Find the equation of the curve passing through the point $\left( {0,\frac{\pi }{4}} \right)$whose differential equation is $\sin x\cos ydx + \cos x\sin ydy = 0$
$\frac{sin x}{cos x}dx = - \frac{sin y}{cos y} dy $
$\int {\tan xdx = - \int {\tan y\,dy} } $
$\log \left( {\sec x} \right) = - \log \left( {\sec y} \right) + \log c$
$\log \left( {\sec x.\sec y} \right) = \log c$
$\sec x.\sec y = c$ ......(i)
When $x = 0,y= \pi /4$
$c = \sqrt 2 $
put the value of c in eq (i)
$\sec x.\sec y = \sqrt 2 $
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