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.

Answer 1

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.