 Solution

ID: past-exam-of-the-algebra-analysis-and-differential-equations-2-course-of-the-university-of-oxford/2011/differential-equations/9/i/solution

\overline{y} (p) = \int_{0}^{\infty} e^{- pt} y (t) d t .

A sufficient condition is that

y

be piecewise continuous on each bounded interval and of exponential order

c

, meaning

∣ y (t) ∣ \leq C e^{c t}

for all sufficiently large

t

. Then the integral is absolutely convergent for

Re p > c

, by comparison with

e^{- (Re p - c) t}

For the constant function, direct integration gives

L [1] (p) = \frac{1}{p} (Re p > 0) .

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