 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2023/paper-2/15/b/i/solution

u^{''} = a u^{'} + b v^{'} = a u^{'} + b c u + b d v .

The original first equation also gives

b v = u^{'} - a u

, so

u^{''} = (a + d) u^{'} + (b c - a d) u .

Therefore the requested coefficients are

α = a + d, β = a d - b c .

They are the trace and determinant of the coefficient matrix. This derivation never divides by

b

and is valid even if

b = 0

. In that degenerate case the scalar second-order equation is a necessary consequence, while the original constraint

u^{'} = a u

must also be retained when reconstructing the coupled system.

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