Linear differential equations sometimes occur in which one or both of the functions $p(t)$ and $g(t)$ for $y^{\text{'}}p(t)y=g(t)$ have jump discontinuities. If $t_0$ is such a point of discontinuity, then it is necessary to solve the equation separately for $t>t_{0}y{t}_0{y}^{\text{'}}{t}_0{y}^{\text{'}}6y=g(t),y(0)=0g(t)=\{\begin{array}{cc}1,& 0t1\\ 0,& t>1\end{array}t$ and $t>t_0.$ Afterward, the two solutions are matched $so$ that $yis$ continuous $at{t}_0$; this $is$ accomplished $by$ a proper choice $of$ the arbitrary constants. The following problem illustrates this situation.

Note that $itis$ impossible also $to$ make $y^{\text{'}}$ continuous $at{t}_0.$

Solve the initial value problem.

$y^{\text{'}}6y=g(t),y(0)=0$

where

$g(t)=\{\begin{array}{cc}1,& 0t1\\ 0,& t>1\end{array}$