For x R, x 0, if y(x) is a differentiable function such that x _1^xy(t)dt = (x+1) _1^xty(t)dt, then y(x) equals (where C is a constant)

JEE Main 2016 Mathematics question

For , , if is a differentiable function such that x\int\limits_1^xy(t)dt = (x+1)\int\limits_1^xty(t)dt, then equals (where is a constant)

Answer: (4)

Differentiate the given identity with respect to , using the product rule on each side:

y dt + x y(x) = t y dt + (x+1)x y(x).

Collecting terms,

(1-t)y(t) dt = .

Differentiate once more:

(1-x)y(x) = 2x y(x)

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Question ID VK-115667