 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2021/paper-1/13/a/ii/solution

y_{p} = A + B cos 2 x

. Substitution gives

y_{p}^{''} + y_{p} = A - 3 B cos 2 x .

Matching the constant and cosine coefficients to

\frac{1}{2} - \frac{1}{2} cos 2 x

gives

A = 1/2

B = 1/6

. Thus

y_{p} (x) = \frac{1}{2} + \frac{1}{6} cos 2 x .

Neither forcing term is resonant with the homogeneous angular frequency one.

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