= Solution
Integration by parts, where the transform boundary term at infinity vanishes, gives
$$
\boxed{\mathcal L[\dot y](p)=p\overline y(p)-y(0).}
$$
For a sufficiently regular function with the required exponential growth bounds, apply this again to $\dot y$ to deduce
$$
\boxed{\mathcal L[\ddot y](p)=p^2\overline y(p)-py(0)-\dot y(0).}
$$
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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