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If cos x (dy)/(dx)-y sin x = 6x, (0 lt x...

If cos `x (dy)/(dx)-y sin x = 6x, (0 lt x lt (pi)/(2))` and `y((pi)/(3))=0`, then `y((pi)/(6))` is equal to :-

A

`- (pi^(2))/(4 sqrt(3))`

B

`- (pi^(2))/(2)`

C

`- (pi^(2))/(2sqrt(3))`

D

`(pi^(2))/(2 sqrt(3))`

Text Solution

Verified by Experts

The correct Answer is:
C
`(dy)/(dx) - y tan x = (6x)/(cos x), x in (0, (pi)/(2))`
If. `= e^(int - tan k) = e^(+ log|cos x|) = cos x`
`y.cos x = int (6x)/(cos x). Cos x dx`, `y cos x = 3 x^(2) + c`
`y ((pi)/(3), 0)`, `0 = 3 ((pi^(2))/(9)) + c`
`C = - (pi^(2))/(3)`, `y cos x = 3 x^(2) - (pi^(2))/(3)`
`y cos (pi)/(6) = 3 ((pi^(2))/(36)) - (pi^(2))/(3) = (pi^(2))/(12) - (pi^(2))/(3) = - (3 pi^(2))/(12) = - (pi^(2))/(4)`, `y = - (pi^(2))/(4) . (2)/(sqrt(3)) = (-pi^(2))/(2sqrt(3))`
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