 Solution

ID: past-exam-of-the-short-options-paper-shared-by-parts-a-and-b-of-the-physics-course-of-the-university-of-oxford/2019/s31/3/a/solution

Put

y^{(0)} = y

and

y^{(1)} = \overset{y}{˙}

. Then

\frac{d}{d t} (y^{(0)} y^{(1)}) = M (0 - k^{2} 1 - 2 γ) (y^{(0)} y^{(1)}) .

The second row is the original damped harmonic oscillator equation and the first row defines the auxiliary velocity. This converts a second-order ordinary differential equation into two first-order equations without changing the required two initial values.

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