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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/2018/paper-1/13/b/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 / 2018 / Paper 1 / 13 / b / Solution by

Codex  0 2026-10-18

Rewrite the linear ordinary differential equation as

y^{'} + y = e^{3 x}

. Multiplication by its integrating factor

e^{x}

gives

(e^{x} y)^{'} = e^{4 x}

. Integrating yields

y (x) = \frac{1}{4} e^{3 x} + C e^{- x} .

The general solution consists of this particular solution and the homogeneous solution

C e^{- x}

.

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