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in Differential equations by (36.3k points)

If f(x) is a function such that x∫(1 – t)f(t) dt for t ∈ [0,x] = ∫tf(t) dt for t ∈ [0,x], f(1) = 1, find f(x).

1 Answer

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Best answer

We have

Applying Newton and Leibnitz formula, we get

(–x2)f'(x) – 2x f(x) + (1 – x)f(x) = 0

(x2)f'(x) + (1 – 3x)f(x) = 0

when x = 1, y = 1, 

then c = –1

Hence, the solution is

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