Consider the electrical circuit governed by the differential equation
\begin{equation*}
L I'' + RI' + \frac{1}{C} I = E'(t).
\end{equation*}
The voltage function, \(E'(t)\text{,}\) might have discontinuities. For example, the voltage in the circuit can be periodically turned on and off. The previous methods that we have used to solve second order linear differential equations may not apply here. However, the Laplace transform, an integral transform, gives a method of solving such equations.
As a second example, let us consider a population of fish that is governed by exponential growth,
\begin{align*}
\frac{dP}{dt} & = kP\\
P(0) & = P_0,
\end{align*}
and suppose that we wish to determine the effects of seasonal fishing. In other words, harvesting will not be continuous. For example, we might only allow fishing at a constant rate \(r\) during the first half of the year,
\begin{equation*}
H(t)
=
\begin{cases}
r & 0 \leq t \leq 1/2, \\
0 & 1/2 \lt t \lt 1, \\
r & 1 \leq t \leq 3/2, \\
0 & 3/2 \lt t \lt 2, \\
& \vdots
\end{cases}
\end{equation*}
Our initial value problem now becomes
\begin{align*}
\frac{dP}{dt} & = kP - H\\
P(0) & = P_0,
\end{align*}
It should be clear that we need some additional tools to analyze differential equations possessing discontinuous terms.
Given an initial value problem
\begin{align*}
ay'' + by' + cy & = g(t)\\
y(0) & = y_0\\
y'(0) & = y_0',
\end{align*}
the idea is to use the Laplace transform to change the differential equation into an equation that can be solved algebraically and then transform the algebraic solution back into a solution of the differential equation. Surprisingly, this method will even work when \(g\) is a discontinuous function, provided the discontinuities are not too bad.