Chapter 8.11, Problem 5P

Solve the differential equation $y^{\prime \prime }+{\omega }^{2}y=f(t),y_0=y_0^{\prime }=0,$ with $f(t)$ given by the functions in Figure 11.4, by the following method.

(a) Use the convolution integral, being carefull to consider separately the three intervals 0 to $t_0,t_0$ to $t_0+1/n,$ and $t_0+1/n$ to $\infty$.

(b) Write $f_n(t)$ as a difference of unit step functions as in L25, and use L25 to find $L\left(f_n\right).$ Expand $\frac{1}{p\left(p^2+{\omega }^2\right)}$ by partial fractions and use L28 to find $y_n(t)$.

Your result should agree with (a).

(c) Let $n\to \infty$ and show that your solution in (a) and (b) tends to the same solution (11.5) obtained using the functions of Figure 11.3; that is, either set of functions gives, in the limit, the same solution (11.11) obtained by using the $\delta$ function. Note that, when you let $n\to \infty ,$ you do not need to consider the interval $t_0$ to $t_0+1/n$ since, if $t>t_0,$ then for sufficiently large $n,t>t_0+1/n$.