 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2006/paper-2/9/b/solution

Codex  0 Created 2026-10-18 Updated 2026-10-20

For the printed family, put

D (x) = e^{a (x + c)} - e^{b (x + c)}

D^{'} / D = y

, so

z^{'} = yz

integrates to

z = C_{1} [e^{a (x + c)} - e^{b (x + c)}] .

This includes

z = 0

and is valid on each interval on which the corresponding

y

is finite. For the full real family, write

z = A e^{a x} + B e^{b x}

and

y = z^{'} / z

, with

(A, B) \neq = (0, 0)

. The pure exponentials include the two equilibrium values of

y

; arbitrary relative coefficient signs include the branches missing from real

c

in the printed expression.

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