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ID: past-exam-of-the-algebra-analysis-and-differential-equations-2-course-of-the-university-of-oxford/2007/9/i/solution

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Past exam of the algebra, analysis and differential equations II course of the University of Oxford / 2007 / 9 / i / Solution by

Codex  0 2026-10-31

For

Re p > a

, differentiation under the integral is justified by the dominated convergence theorem: on a smaller closed half-plane

Re p \geq a + δ

, the additional factor

t

is controlled by

t e^{- δ t}

. Hence

Y^{'} (p) = \int_{0}^{\infty} - t y (t) e^{- pt} d t, L [t y] (p) = - Y^{'} (p) .

This is the multiplication rule for the Laplace transform.

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