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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))/(4sqrt(3))`

B

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

C

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

D

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

Text Solution

Verified by Experts

The correct Answer is:
C
`(dy)/(dx)-y tan x =(6x)/(cos x)`
(linear in y)
`I.F. = e^(int-tan xdx)=e^(+ln cos x)=cos x`
`y.cos x=int(6x)/(cos x).cos x dx + C`
`y.cos x=3x^(2)+C`
`therefore y(pi//3)=0` Put `x = (pi)/(3)`. & y = 0
`0 = (3pi^(2))/(9)+C to C = - (pi^(2))/(3)`
`therefore y cos x = 3x^(2)-(pi^(2))/(3)`
`y((pi)/(6))= ?` Put `x = (pi)/(6)`
`y.(sqrt(3))/(2)=3.(pi^(2))/(36)-(pi^(2))/(3)=(-pi^(2))/(4)`
`y = (-pi^(2))/(2sqrt(3))`
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