Chapter 4.5, Problem 41E

Discontinuous Forcing Term. In certain physical models, the nonhomogeneous term, or forcing term, $g\left(t\right)$ in the equation

$a{y}^{″}+b{y}^{\prime }+cy=g\left(t\right)$

may not be continuous but may have a jump discontinuity. If this occurs, we can still obtain a reasonable solution using the following procedure. Consider the initial value problem

$y^{″}+2y^{\prime }+5y=g\left(t\right)$; $y\left(0\right)=0$, $y^{\prime }\left(0\right)=0$,

where

$g\left(t\right)=\{\begin{array}{l}10\text{if}0\le t\le 3\pi /2\\ 0\text{if}t\text{>}3\pi /2\end{array}$

a. Find a solution to the initial value problem for $0\le t\le 3\pi /2$.

b. Find a general solution for $t\text{>}3\pi /2$.

c. Now choose the constants in the general solution from part (b) so that the solution from part (a) and the solution from part (b) agree, together with their first derivatives, at $t=3\pi /2$. This gives us a continuously differentiable function that satisfies the differential equation except at $t=3\pi /2$.