 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-341/2/a/solution

Codex  0 Created 2026-09-28 Updated 2026-10-05

c_{1}, c_{2}

ℓ_{1} (s) = \frac{s - c _{2}}{c _{1} - c _{2}}, ℓ_{2} (s) = \frac{s - c _{1}}{c _{2} - c _{1}} .

Y_{i} = y_{n} + h \sum_{j = 1}^{2} a_{ij} f (Y_{j}), y_{n + 1} = y_{n} + h \sum_{j = 1}^{2} b_{j} f (Y_{j}),

where

a_{ij} = \int_{0}^{c_{i}} ℓ_{j} (s) d s

and

b_{j} = \int_{0}^{1} ℓ_{j} (s) d s

. Explicitly,

A = \frac{c _{1} ( c _{1} /2 - c _{2} )}{c _{1} - c _{2}} - \frac{c _{2}^{2}}{2 ( c _{1} - c _{2} )} \frac{c _{1}^{2}}{2 ( c _{1} - c _{2} )} \frac{c _{2} ( c _{2} /2 - c _{1} )}{c _{2} - c _{1}}, b = \frac{1/2 - c _{2}}{c _{1} - c _{2}} \frac{1/2 - c _{1}}{c _{2} - c _{1}} .

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