$(10$ points$)$ A $1$ kg mass is attached to a spring with spring constant $k=144\frac{N}{m}.$ At time $t=0,$ the mass is pulled down $60$ cm and is then set into motion with an initial downward velocity of $10c\frac{m}{s}.$

$($a$)$ Assuming that the displacement $y(t)$ from the spring$-$mass equilibrium is measured positive in the downward direction, apply Hooke's law and Newton's first and second law of motion to model the differential equation for $y(t).$

$($b$)$ Show that the Laplace transform satisfies the formulas

$[$:$L{f}^{\text{'}\text{'}}(t)$