 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2016/ib/paper-4/14a/b/solution

Codex  0 Created 2026-09-24 Updated 2026-10-06

Write

F (p) = L [f] (p)

. Assume

f

and

f^{'}

have right limits at zero and sufficient regularity and exponential-order bounds for the following integration by parts; take

Re p

large enough to kill the boundary terms at infinity. First,

L [f^{'}] (p) = [e^{- pt} f (t)]_{0}^{\infty} + pF (p) = pF (p) - f (0 +) .

Apply the same calculation to

f^{'}

L [f^{''}] (p) = p^{2} F (p) - p f (0 +) - f^{'} (0 +) .

This is the Laplace transform of a derivative formula with both initial-data terms retained.

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