6.8 Integral Equations

An integral equation is an equation in which the unknown function, call it \(y(t)\), occurs in the integrand of an integral, and may also occur outside it. When the integral is a convolution, the convolution theorem turns the equation into an algebraic one for \(Y(s)\).

Example 6.8.1. Solve the equation \(y(t) = t + \int ^t_0 y(x)\sin (t -x)\,dx\).

Solution. The equation is \(y = t + y*\sin t\). By the convolution theorem, \(\mathcal {L}\{y*\sin t\} = Y(s)\cdot \frac {1}{s^2 + 1}\), so transforming, \[Y(s) =\frac {1}{s^2} + \frac {Y(s)}{s^2 + 1},\qquad \frac {s^2}{s^2 + 1}\,Y(s) = \frac {1}{s^2} ,\] and \[Y(s) = \frac {s^2 + 1}{s^4} = \frac {1}{s^2} + \frac {1}{s^4},\qquad y(t) = t + \frac {1}{6}t^3 .\]

Example 6.8.2. Solve \(y'(t) + 2y(t) + \int _0^t y(\tau )\,d\tau = 1\), \(y(0) = 0\).

Solution. Transforming, with Theorem 6.3.1 for the integral, \[sY + 2Y + \frac {Y}{s} = \frac 1s,\qquad Y(s^2 + 2s + 1) = 1,\qquad Y = \frac {1}{(s + 1)^2},\] so \(y(t) = te^{-t}\).

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