MATH319 Slides 90 Example of Laplace transform of differential equation 92 Solving MIMO by Laplace transforms

91 Example of Laplace transform of differential equation, concluded

substituting this into

{\cal L}(y^{\prime\prime\prime})+2{\cal L}(y^{\prime\prime})-{\cal L}(y^{% \prime})+{\cal L}(y)={\cal L}(u)

gives

(s^{3}+2s^{2}-s+1)\hat{y}(s)-7s^{2}-13s+4=\hat{u}(s).

Remarks (i) The characteristic equation in frame 88 is the same as we get from frames 19 and 20.

(ii) We often take $s$ such that $\Re s\geq 0$ , so $s$ is in the right half plane, and denote the points on the imaginary axis by $s=i\omega$ , where $\omega\in{\bf R}$ is the angular frequency.