 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2024/paper-1/3/i/solution

For an exponential trial

y = e^{r x}

r^{2} + 2 a r + a^{2} + b^{2} = (r + a)^{2} + b^{2} = 0.

Its distinct complex roots are

r = - a \pm ib

. Thus the general real solution of the homogeneous second-order linear differential equation is

y_{h} (x) = e^{- a x} [C_{1} cos (b x) + C_{2} sin (b x)] .

The two displayed solutions have nonzero Wronskian

b e^{- 2 a x}

and hence linear independence. The same formula permits complex solutions when

C_{1}, C_{2}

are complex.

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