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Question

For xR, x0 if y(x) is a differentiable function such that xx1y(t) dt=(x+1)x1t y(t) dt, then y(x) equals:

A
Cx3 e1x
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B
Cx2 e1x
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C
Cx3 e1x
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D
Cx e1x
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Solution

The correct option is C Cx3 e1x
xx1y(t) dt=(x+1)x1t y(t) dt
Differentiating w.r.t x, we get
x1y(t) dt+x y(x)=x1t y(t) dt+(x+1)x y(x)
x2 y(x)=x1(y(t)t y(t)) dt
Again differentiating w.r.t x, we get
2x y(x)+x2dydx=y(x)x y(x)
dydx=13xx2 y
dyy=13xx2 dx
logy=1x3logx+logC
logy=1xlogx3+logC
logy x3C=1x
y=Cx3e1x

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