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If the function y(x) satisfies the integral equation ∫[a to x] t*y(t) dt = x² + y(x), then y expressed as a function of x is:

1. y = 2 - (2 + a²)e^((x²-a²)/2)
2. y = 1 - (2 + a²)e^((x²-a²)/2)
3. y = 2 - (1 + a²)e^((x²-a²)/2)
4. none

Correct answer: y = 2 - (2 + a²)e^((x²-a²)/2)

Solution

Differentiating gives y' - x*y = -2x, whose solution with y(a) = -a² (from the equation at x=a) is y = 2 - (2 + a²)e^((x²-a²)/2).

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