 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2006/paper-1/10/c/solution

Codex  0 Created 2026-10-18 Updated 2026-10-20

(D - 1)^{2}

, where

D = d / d x

y = e^{x} v

; the product rule reduces the equation to

v^{''} = 1 + x

. Twice integrating gives

v = x^{2} /2 + x^{3} /6 + A x + B

B = A = 0

, so

y (x) = e^{x} (\frac{x ^{2}}{2} + \frac{x ^{3}}{6}) .

This substitution removes the repeated characteristic root before finding a particular solution.

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