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 the integral). Integral equations can be written as a convolution can be solved by the Laplace transformation.
Solve the equation \(\hspace {0.5cm}\displaystyle {y(t) = t + \int ^t_0 y(x)\sin (t -x)dx}\)
Solution.
From the equation, we see that \(\hspace {0.4cm} y(t) = t + y(t)*\sin t\). By the convolution theorem \[\mathcal {L}\{f(t)*g(t)\} = F(s)\hspace {0.1cm} G(s)\] where \(F(s) = \mathcal {L}\{f\},\hspace {0.4cm} G(s) = \mathcal {L}\{g\}\). Here, \(f(t) = y(t),\hspace {0.4cm} g(t) = \sin t\)
\begin {align*} \mathcal {L}\{y(t)*\sin t\} & = \mathcal {L}\{y(t)\}\cdot \mathcal {L}\{\sin t\}\\ & = Y(s)\cdot \frac {1}{s^2 + 1} \end {align*}
So \(\hspace {0.5cm}\displaystyle {\mathcal {L}\{y(t)\} = \mathcal {L}\{t\} + \mathcal {L}\{y*\sin t\}}\)
\[Y(s) =\frac {1}{s^2} + Y(s)\cdot \frac {1}{s^2 + 1}\]
\[\Bigg (1 - \frac {1}{s^2 + 1}\Bigg ) Y(s) = \frac {1}{s^2}\]
\begin {align*} Y(s) & = \frac {1}{s^2}\cdot \frac {s^2 + 1}{s2}\\ Y(s) & = \frac {s^2 + 1}{s^4} = \frac {1}{s^2} + \frac {1}{s^4} \end {align*}
\[\therefore \hspace {0.5cm} y(t) = t + \frac {1}{6}t^3\]
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