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Question
Find the equation of the curve passing through the point `(0,pi/4)`, whose differential equation is sin x cos y dx + cos x sin y dy = 0.
Solution
We have sin x cos y dx + cos x sin y dy = 0
⇒ `sin x/cos x dx + siny/cos y dy = 0`
Integrating, `- int (- sin x)/cos x dx - int (- sin y)/ cos y dy = ` constant
⇒ - log |cos x| - log |cos y| = - log |C|
⇒ - log |cos x cos y| = - log |C|
⇒ cos x cos y = C .....(1)
Since the curve passes through `(0, pi/4)`
∴ `cos 0 cos pi/4 = C`
⇒ `(1) (1/sqrt2) = C`
⇒ `C = 1/sqrt 2`
Putting `C = 1/sqrt 2` in (1)
Cos x cos y = `1/sqrt2`
⇒ `cos y = sec x/sqrt2`
Which is the required equation for the curve.
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