The solution of dy = cos x `(2 - y " cosec " x) dx` , where ` y = sqrt(2) , " when " x = pi//4 ` is

Options

A. ` y = sin x +1/2 "cosec"x`
B. ` y = tan (x//2)+cot(x//2)`
C. `y = (1//sqrt(2))sec (x//2)+sqrt(2)cos (x//2)`
D. None of the above

Answer

b

Explanation

Given , `(dy)/(dx) = 2 cos x - y cos x "cosec " x`
` rArr (dy)/(dx) + y cot x = 2 cos x `
which is linear differential equation .
` :. IF = e^(int cot x dx ) = e ^(log (sin x) ) = sin x `
` :. " Required Solution is " y * sin x = int 2 cos x sin x dx +c`
` rArr y sin x = int sin 2 x dx +C` ,
` rArr y sin x = (- cos 2 x)/2 +C `
Given at ` x = pi/4 , y = sqrt(2)`
` :. sqrt(2) sin . pi/4 = (- cos 2(pi //4))/2 +C`
` rArr C = 1`
` :. y sin x = -1/2 cos 2 x +1`
` rArr y = -1/2 * (cos 2x)/(sinx) + " cosec " x `
` rArr y = 1/(2sin x ) (1-2 sin^(2)x) + " cosec " x`
` rArr y = 1/2 " cosec " x + sin x `