Answer
$$
y=\{-0.0232 \sin 8.97 t+0.333 \cos 8.97 t+0.0520 \sin 4 t\} \mathrm{ft}
$$
Work Step by Step
$$
\begin{gathered}
y=A \sin \omega_n t+B \cos \omega_n t+\frac{\delta_0}{1-\left(\frac{\omega_2}{\omega_0}\right)^2} \sin \omega_0 t \\
v=\dot{y}=A \omega_n \cos \omega_n t-B \omega_n \sin \omega_n t+\frac{\delta_0 \omega_0}{1-\left(\frac{\omega_0}{\omega_n}\right)^2} \cos \omega_0 t
\end{gathered}
$$
The initial condition when $t=0, y=y_0$, and $v=v_0$ is
$$
\begin{aligned}
& y_0=0+B+0 \quad B=y_0 \\
& v_0=A \omega_n-0+\frac{\delta_0 \omega_0}{1-\left(\frac{\omega_0}{\omega_n}\right)^2} \quad A=\frac{v_0}{\omega_n}-\frac{\delta_0 \omega_0}{\omega_n-\frac{\omega_0^2}{\omega_n}}
\end{aligned}
$$
Thus,
$$
\begin{aligned}
& y=\left(\frac{v_0}{\omega_n}-\frac{\delta_0 \omega_0}{\omega_n-\frac{\omega_0^2}{\omega_n}}\right) \sin \omega_n t+y_0 \cos \omega_n t+\frac{\delta_0}{1-\left(\frac{\omega_n}{\omega_n}\right)^2} \sin \omega_0 t \\
& \omega_n=\sqrt{\frac{k}{m}}=\sqrt{\frac{10}{4 / 32.2}}=8.972 \\
& \frac{\delta_0}{1-\left(\frac{\omega_0}{\omega_n}\right)^2}=\frac{0.5 / 12}{1-\left(\frac{4}{8.972}\right)^2}=0.0520 \\
& \frac{v_0}{\omega_n}-\frac{\delta_0 \omega_0}{\omega_n-\frac{\omega_0^2}{\omega_0}}=0-\frac{(0.5 / 12) 4}{8.972-\frac{4^2}{8.972}}=-0.0232 \\
& y=(-0.0232 \sin 8.97 t+0.333 \cos 8.97 t+0.0520 \sin 4 t) \mathrm{ft}
\end{aligned}
$$
