Question

1 {d^2theta }/{dt^2}+{b/l}*{d\theta }/{dt}+{k/l}\theta =0

y1(t) = theta(t) and y2(t) = theta'(t) , show that equation can be written as a system of first order ODEs involving y1 and y2.
Summarise the system in the matrix equation y'= M y
2 Show that the eigenvalues of M

Answer 1

(a) Given the ODE

`(d^2\theta)/(dt^2) + \frac bI (d\theta)/(dt) + \frac kI \theta = 0`

and substituting
`y_1 = \theta` and
`y_2=\theta'`, it follows that
`{y_1}' = y_2` and
`{y_2}' = \theta''`, so we get the system of ODEs

`\begin{cases} (dy_2)/(dt) + \frac bI y_2 + \frac kI y_1 = 0 \\\\ (dy_1)/(dt) = y_2 \end{cases}`

which we can write in matrix form as

`\begin{bmatrix}y_1\\y_2\end{bmatrix}' = \begin{bmatrix}0 & 1 \\\\ -\frac kI & -\frac bI\end{bmatrix} \begin{bmatrix}y_1 \\ y_2\end{bmatrix}`

(b) Compute the eigenvalues for the coefficient matrix.

`\det\begin{bmatrix} -\lambda & 1 \\\\ - \frac kI & -\frac bI - \lambda \end{bmatrix} = \lambda^2 + \frac bI \lambda + \frac kI = 0 \\\\ \implies \lambda = \frac{-\frac bI \pm \sqrt{\left(\frac bI\right)^2 - \frac{4k}I}}2 = (-b \pm √(b^2-4kI))/(2I)`