 Solution

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

Integration by parts, where the transform boundary term at infinity vanishes, gives

L [\overset{y}{˙}] (p) = p \overline{y} (p) - y (0) .

For a sufficiently regular function with the required exponential growth bounds, apply this again to

\overset{y}{˙}

to deduce

L [\overset{y}{¨}] (p) = p^{2} \overline{y} (p) - p y (0) - \overset{y}{˙} (0) .

These identities hold in a right half-plane where the displayed transforms converge; the regular solutions obtained below satisfy those requirements.

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