 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2022/paper-2/7/b/ii/solution

Let

z = ω_{1}^{2}

, so

ω_{2}^{2} = 4 z

. The sum and product of roots of the characteristic polynomial in (i) give

5 z = \frac{2 ( m _{2} + 1 )}{m _{2}}, 4 z^{2} = \frac{3}{m _{2}} .

Eliminating

z

yields

16 (m_{2} + 1)^{2} = 75 m_{2}, 16 m_{2}^{2} - 43 m_{2} + 16 = 0.

m_{2} = \frac{43 \pm 5 33}{32} ≃ 0.446162086 or 2.241337914.

The two values are reciprocals. Both meet the requested frequency ratio; substituting either into the expressions in (i) gives

ω_{2} / ω_{1} = 2

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