Chapter 5.8, Problem 23P

Consider the equation

$\varphi \left(t\right)+{\displaystyle \underset{0}{\overset{t}{\int }}k\left(t-\xi \right)}\varphi \left(\xi \right)d\xi =f\left(t\right)$,

in which $f$ and $k$ are known functions, and $\varphi$ is to be determined. Since the unknown function $\varphi$ appears under an integral sign, the given equation is called an integral equation; in particular, it belongs to a class of integral equations known as Volterra integral equations. Take the Laplace transform of the given integral equation and obtain an expression for $L\left\{\varphi \left(t\right)\right\}$ in terms of the transforms $L\left\{f\left(t\right)\right\}$ and $L\left\{k\left(t\right)\right\}$ of the given functions $f$ and $k$. The inverse transform of $L\left\{\varphi \left(t\right)\right\}$ is the solution of the original integral equation.