Chapter 2.4, Problem 27P

Discontinuous Coefficients. Linear differential equationssometimes occur in which one or both of the functions $p$ and $g$ have jump discontinuities. If $t_0$ is such a point of discontinuity, then it is necessary to solve the equation separately for $t<t_0$ and $t>t_0$. Afterward, the two solutions are matched sothat $y$ is continuous at $t_0$. This is accomplished by a properchoice of the arbitrary constants. Problems $27$ and $28$ illustrate this situation. Note in each case that it is impossible to make $y\text{'}$ continuous at $t_0$: explain why, just from examining the differential equations.

Solve the initial value problem

$y\text{'}+2y=g\left(t\right),y\left(0\right)=0$,

Where

$g\left(t\right)=\{\begin{array}{l}1,0\le t\le 1,\\ 0,t>1.\end{array}$