Question

Use the Laplace transform to solve the given integral equation or integrodifferential equation.

$\mathbf{f}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\mathbf{cost}\mathbf{+}\underset{\mathbf{0}}{\overset{\mathbf{t}}{\mathbf{\int }}}{\mathbf{e}}^{\mathbf{-}\mathbf{\tau }}\mathbf{f}\mathbf{(}\mathbf{t}\mathbf{-}\mathbf{\tau }\mathbf{)}\mathbf{d\tau }$

Short answer

The laplace transformation is $f\left(t\right)=\frac{2}{5}t+\frac{8}{5\sqrt{5}}\sin \sqrt{5}t$

Step-by-step solution

Using Convolution Theorem;

Here we will start solving the question using the convolution theorem

$L\left\{\underset{0}{\overset{t}{\int }}f\left(\tau \right)g\left(t-\tau \right)d\tau \right\}=L\left\{f\left(t\right)\right\}L\left\{g\left(t\right)\right\}$

Using Laplace Transformation formulas;

Take Laplace transformation of both the sides;

$\begin{array}{c}L\left\{f\left(t\right)\right\}=L\left\{2t\right\}-4L\left\{\underset{0}{\overset{t}{\int }}\sin \tau f\left(t-\tau \right)d\tau \right\}\\ F\left(s\right)=L\left\{2t\right\}-4L\left\{\sin t\right\}L\left\{f\left(\tau \right)\right\}\\ F\left(s\right)=\frac{2\left(s^2+1\right)}{s^2(s^2+5)}\\ \\ F\left(s\right)=\frac{A}{s^2}+\frac{B}{s^2+5}\\ L^{-1}\left\{F\left(s\right)\right\}=\frac{2}{5}{L}^{-1}\left\{\frac{1}{s^2}\right\}+\frac{8}{5}{L}^{-1}\left\{\frac{1}{s^2+5}\right\}\\ f\left(t\right)=\frac{2}{5}t+\frac{8}{5\sqrt{5}}\sin \sqrt{5}t\end{array}$

Hence, the final answer is$f\left(t\right)=\frac{2}{5}t+\frac{8}{5\sqrt{5}}\sin \sqrt{5}t$