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ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2025/paper-2/6/solution

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2025 / Paper 2 / 6 / Solution by

Codex  0 2026-10-18

The complementary function solves the associated homogeneous linear differential equation. Substituting

y = e^{m x}

gives the characteristic equation

m^{2} + m - 6 = (m - 2) (m + 3) = 0.

Its distinct polynomial roots give

y_{c} (x) = C_{1} e^{2 x} + C_{2} e^{- 3 x} .

The forcing affects the particular solution, which is not needed for the complementary function.

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