linear ODE with discontinuous nonhomogeneous term

Question

In Boyce's Elementary Differential Equations and Boundary Value Problems(global edition, 2017) pp 264-5 it says that if $y$ is a solution of 2nd order ODE $y''+p(t)y'+q(t)y=g(t)$ where $p, q$ are continuous and $g$ is only piecewise continuous on an interval, $y$ and $y'$ are continuous on the interval but $y''$ has jump discontinuities at the same point as $g$. Furthermore it says the same about higher order linear ODEs. The book doesn't have any proof for this, so I would like to know how to prove it.

Answer 1

This case is not the basic case of an ODE where all the components are continuous functions. However, under the given assumptions the integration interval has a finite subdivision so that $g$ is continuous and bounded over each segment and thus has a solution there in the strong sense.

The question now is how to define a generalized solution over the full interval. The minimal condition is of course that the solution function be continuous. Next up that it has a continuous derivative. Now the jump discontinuities on the right side only have one place to occur also on the left side, and that is in $y''$.

One could also put this in another way, let $G$ be a continuous integral function of $g$. Then set $u(x)=y'(x)-G(x)$ so that then one obtains in $y,u$ the first order system \begin{align} y'&=u+G\\ u'&=y''-G'=-py'-qy\\ &=-p(u+G)-qy \end{align} This now satisfies the continuity assumptions for a linear system over the full integration interval. Again, in the result the jumps of $g$ are replicated as jumps in $y''$.