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If y= y(x) is the solution of the differ...

If y= y(x) is the solution of the differential equation
`dy/dx=(tan x-y) sec^(2)x, x in(-pi/2,pi/2),` such that y (0)=0,
than `y(-pi/4)` is equal to

A

e-2

B

`2+1/e`

C

`1/e-2`

D

`1/2-e`

Text Solution

Verified by Experts

The correct Answer is:
A
`(dy)/(dx) = (tanx-y) sec^(2)x rArr (dy)/(dx) + y sec^(2)x = tan x sec^(2)x`
It is a L.D.E.
I.F = `e^(int sec^(2)x dx) = e^(tan x)`
`y(e^(tanx)) = inte^(tanx).tan x.sec^(2)x dx`
Put tanx =t
`y(e^(tanx)) = int e^(t).tdt`
`y(e^(tanx)) =tanx.e^(tanx)-e^(tanx)+c`
`y(x) =tan x-1+ce^(-tanx)`
Now y(0)=0
0=0-1+c
c=1
y(x) = tan x -1 `+e^(-tan x)`
Now `y(-pi/4) rArr y(-pi/4) =-1+1+e^(+1) rArr y(-pi/4) =e-2`
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