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If (dy)/(dx)+3/(cos^(2)x)y=1/(cos^(2)x),...

If `(dy)/(dx)+3/(cos^(2)x)y=1/(cos^(2)x),xin((-pi)/3,(pi)/3),` and `y((pi)/4)=4/3`, then `y(-(pi)/4)` equals `1/3+e^(k)`. The value of k is __________.

Text Solution

Verified by Experts

The correct Answer is:
6
`I.F=e^(int3sec^(2)xdx)=e^(3tanx)`
`yxxe^(3tanx)=intsec^(2)xe^(3tanx)dx`
`yxxe^(3tanx)=1/3e^(3tanx)+c`
`4/3xxe^(3)=1/3e^(3)+c`
`c=e^(3)`
`y=1/3+(e^(3))/(e^(3tanx))`
`y(-(pi)/4)=1/3+e^(6)`
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