Convolution

An integral product whose transform is the product of the transforms, used to invert awkward expressions and to solve Volterra integral equations.

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4 exercises

Let $f(t) = \cos t$ for $t \geq 0$ , and $g(t) = e^{-t}$ . Compute $f * g$ .

Compute ${\mathcal{L}}^{-1} \left\{ \frac{5}{s^4+s^2} \right\}$ using convolution.

Solve $x''+x = \sin t$ , $x(0) = 0$ , $x'(0)=0$ using convolution.

Solve $x'''+x' = f(t)$ , $x(0) = 0$ , $x'(0)=0$ , $x''(0)=0$ using convolution. Write the result as a definite integral.