 Solution

ID: past-exam-of-the-mathematical-methods-i-course-of-the-university-of-oxford/2019-resit/10/a/solution

e^{2 x}

(e^{2 x} y)^{'} = 1 + e^{2 x} (x^{2} + 2 x)

x^{2} /2 + x /2 - 1/4

, while the resonant exponential forcing contributes

x e^{- 2 x}

. Consequently

y = (x + C) e^{- 2 x} + \frac{x ^{2} + x}{2} - \frac{1}{4} .

Substitution gives

y^{'} + 2 y = e^{- 2 x} + x^{2} + 2 x

, and the free multiple of

e^{- 2 x}

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