(Solved): Consider the initial value problem y+16y=g(t),y(0)=0,y(0)=0, where g(t)={t0if0t ...

Consider the initial value problem $$y^{??}+16y=g(t),y(0)=0,y^{?}(0)=0,$$ where $$g(t)=\{\begin{array}{ll}t& \text{ if }0?t<2\\ 0& \text{ if }2?t<?\end{array}$$ a. Take the Laplace transform of both sides of the given differential equation to create the corresponding algebraic equation. Denote the Laplace transform of $$y(t)$$ by $$Y(s)$$. Do not move any terms from one side of the equation to the other (until you get to part (b) below). $$=$$ help (formulas) b. Solve your equation for $$Y(s)$$. $$Y(s)=\mathcal{L}\{y(t)\}=$$ c. Take the inverse Laplace transform of both sides of the previous equation to solve for $$y(t)$$. If necessary, use $$h(t)$$ to denote the Heaviside function $$h(t)=\{\begin{array}{ll}0& \text{ if }t<0\\ 1& \text{ if }0?t\end{array}$$. $$y(t)=$$