Consider the initial value problem for $0 < t < \infty$: $$ay'' + by' + cy = f(t), \quad \quad y(0) = 0, \quad y'(0) = 0,$$ where $a, b, c$ are constants and $f(t)$ is a known function. We can view this problem as defining a linear system, where $f(t)$ is a known input and the corresponding solution $y(t)$ is the output. Laplace transforms of the input and output functions satisfy the multiplicative relation $$Y(s) = \Phi(s) F(s),$$ where $\Phi(s) = \displaystyle\frac{1}{as^2+bs+c}$ is the system transfer function.

Suppose an input $f(t) = 8t$, when applied to the linear system above, produces the output $y(t) = 2\!\left(e^{-3t}-1\right)+t\!\left(e^{-3t}+5\right), \ t\geq 0.$

1. Find $Y(s) = {\mathcal L}\left\lbrace y(t) \right\rbrace$ and $F(s) = {\mathcal L}\left\lbrace f(t) \right\rbrace$.
   
   $\displaystyle Y(s) =$
   
   $\displaystyle F(s) =$
   
   
2. Use your answer to part (a) to find the system transfer function, $\Phi(s)$.
   
   $\displaystyle \Phi(s) =$
   
   
3. What will be the output if a Heaviside unit step input $f(t) = h(t)$ is applied to the system?
   
   New $\displaystyle y(t) =$ help (formulas)