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A function y(x) satisfies the integro-differential equation y'(x) = y(x) + integral from 0 to 1 of y(x) dx, with y(0) = 1. Find the value of y evaluated at x = ln(11 - 3e) / 2.

1. [2*sqrt(11-3e) - (e-1)] / (3-e)
2. (11 - 3e) / (3 - e)
3. 2 / (3 - e)
4. sqrt(11 - 3e)

Correct answer: [2*sqrt(11-3e) - (e-1)] / (3-e)

Solution

Setting lambda = integral₀¹ y dx, the ODE becomes y' = y + lambda. General solution: y = A*e^x - lambda. From y(0)=1: A = 1+lambda. Self-consistency: lambda = integral₀¹ [(1+lambda)e^x - lambda] dx = (1+lambda)(e-1) - lambda. Solving: lambda(3-e) = e-1, lambda = (e-1)/(3-e). Then A = 2/(3-e). At x = ln(11-3e)/2, e^x = sqrt(11-3e). y = [2*sqrt(11-3e) - (e-1)] / (3-e).

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