Writing a homogeneous oscillator DE as a first-order vector DE

Question

Trying to understand the connection between oscillations and vector DEs, how would I write the homogenous oscillator DE $$y’’+ 2\lambda y’ + w_0^2y = 0$$ as a first-order vector DE?

Any help would be greatly appreciated.

Answer 1

Set

$z = y', \; z' = y''; \tag 1$

then the equation

$y'' + 2\lambda y' + \omega_0^2 y = 0 \tag 2$

becomes

$z' + 2\lambda z + \omega_0^2 y = 0, \tag 3$

that is,

$ z' = -2\lambda z - \omega_0^2 y; \tag 4$

if we now define

$\mathbf r = \begin{pmatrix} y \\ z \end{pmatrix}, \tag 5$

then

$\mathbf r' = \begin{pmatrix} y' \\ z' \end{pmatrix} = \begin{pmatrix} z \\ -2\lambda z - \omega_0^2 y \end{pmatrix} = \begin{bmatrix} 0 & 1 \\ -\omega_0^2 & -2\lambda \end{bmatrix} \begin{pmatrix} y \\ z \end{pmatrix} = \begin{bmatrix} 0 & 1 \\ -\omega_0^2 & -2\lambda \end{bmatrix} \mathbf r. \tag 6$

We have thus reduced the second-order scalar equation (2) to the first-order linear vector system

$\mathbf r^\prime = \begin{bmatrix} 0 & 1 \\ -\omega_0^2 & -2\lambda \end{bmatrix} \mathbf r, \tag 7$

as per request.