Chapter 5.2, Problem 27P

In each of Problems $23$ through $27$, find the Laplace transformation $Y\left(s\right)=L\left\{y\right\}$ of the solution of the given initial value problem. A method of determining the inverse transform is developed in Section $5.5$.

A damped oscillator with mass $m$, damping constant $\gamma$, and a spring constant $k$ is subjected to an external force $F\left(t\right)=F_{0}t$ over the time interval $0\le t\le T$. The external force is then removed at time $T$. Find the Laplace transform of the displacement $y\left(t\right)$ of the mass, assuming that the oscillator is initially in the equilibrium state.