Past exam of the algebra, analysis and differential equations II course of the University of Oxford / 2011 / Differential Equations / 9 / iii / Solution

Integration by parts, where the transform boundary term at infinity vanishes, givesFor a sufficiently regular function with the required exponential growth bounds, apply this again to $\dot{y}$ to deduceThese identities hold in a right half-plane where the displayed transforms converge; the regular solutions obtained below satisfy those requirements.