Past exam of the mathematics course of the University of Cambridge / 2018 / ib / Paper 3 / 7A / b

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- Solution b

Solution

The Green-function representation gives

y (x) = \int_{0}^{1} G (x; ξ) e^{2 ξ} d ξ = e^{2 x} [(x - 1) \int_{0}^{x} ξ d ξ + x \int_{x}^{1} (ξ - 1) d ξ] .

The bracket simplifies to

x (x - 1) /2

, so

y (x) = \frac{1}{2} x (x - 1) e^{2 x} .

It vanishes at both endpoints, and applying

(D - 2)^{2}

returns

e^{2 x}

.

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