Chapter 8.11, Problem 6P

(a) Let a mechanical or electrical system be described by the differential equation $A{y}^{\prime \prime }+B{y}^{\prime }+Cy=f(t),y_0=y_0^{\prime }=0.$ As in Problem 10.16b, write the solution as a convolution (assume $a\ne b$ ). Let $f(t)$ be one of the functions in Figure 11.4 and Problem 5. Find $y$ and then let $n\to \infty$.

(b) Solve (a) with $f(t)=\delta \left(t-t_0\right);$ your result should be the same as in (a).

(c) The solution $y$ as found in (a) and (b) is called the response of the system to a unit impulse. Show that the response of a system to a unit impulse at $t_0=0$ is the inverse Laplace transform of the transfer function.