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If y = y ( x ) is the solutio...

If ` y = y ( x ) ` is the solution of differential equation ` sin y (dy ) /(dx ) - cos y = e ^ ( - x ) ` such that ` y ( 0 ) = ( pi ) /(2) ` then ` y (A) ` is equal to

A

` sin ^( -1 ) "" (1 ) /(e ) `

B

` cos ^( -1) "" ( 1 )/(e ) `

C

` - cos ^( -1) "" (1 ) / (e ) `

D

` cos ^( -1) ( - (1 ) /(e ) ) `

Text Solution

Verified by Experts

The correct Answer is:
D
` sin y (dy)/(dx) - cos y = e^(-x)`
Put ` - cos y = t " " rArr " " sin y (dy)/(dx) = (dt)/(dx) " "rArr " " (dy)/(dx) + t = e^(-x)`
`IF = e^(int dx) = e^(x) rArr " " t.e^(x) int e^(-x) e^(x) dx + C " " rArr " " - cos y.e^(x) = x + C`
`therefore " " y (0) = (pi)/(2) 0 = 0 + C = 0 , cos y = - xe^(-x) rArr " " y(1) = cos^(-1)(-(1)/(e))`
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