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Let y=f(x) satisfies the equation f(x)...

Let `y=f(x)` satisfies the equation
`f(x) = (e^(-x)+e^(x))cosx-2x-int_(0)^(x)(x-t)f^(')(t)dt`
y satisfies the differential equation

A

`(dy)/(dx)+y=e^(x)(cosx-sinx)-e^(-x)(cosx+sinx)`

B

`(dy)/(dx)-y=e^(x)(cosx-sinx)-e^(-x)(cosx+sinx)`

C

`(dy)/(dx)+y=e^(x)(cosx+sinx)-e^(-x)(cosx-sinx)`

D

`(dy)/(dx)-y=e^(x)(cosx-sinx)+e^(-x)(cosx-sinx)`

Text Solution

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The correct Answer is:
A
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Knowledge Check

  • Let y=f(x) satisfies the equation f(x) = (e^(-x)+e^(x))cosx-2x+int_(0)^(x)(x-t)f^(')(t)dt The value of f(0)+f^(')(0) equal

    A
    `-1`
    B
    0
    C
    1
    D
    1

  • Let y=f(x) satisfies the equation f(x)=(e^(-x)+e^(x)) cosx-2x-int_(0)^(x)(x-t)f'(t)dt. The value of f'(0)+f''(0)equals to

    A
    -1
    B
    2
    C
    1
    D
    0

  • If int_(0)^(x)f(t)dt=e^(x)-ae^(2x)int_(0)^(1)f(t)e^(-t)dt , then

    A
    `a=(1)/(3-2e)`
    B
    `f(x)=e^(x)-2e^(2x)`
    C
    `a=(1)/(e)`
    D
    `f(x)=e^(x)-e^(-x)`

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