Convert vector differential equation's order

Question

Here is a 2nd order vector differential equation: $$\overrightarrow{Y}''= \begin{pmatrix}a & b \\c & d \end{pmatrix} \overrightarrow{Y}$$ Don't work it out, but write it as a vector differential eqn in $1$st order in higher dimensions.

I am not sure where to begin.

How does one convert from 2nd to 1st order? Hints are appreciated.

Answer 1

HINT

Here's a simple example. If you have the second order differential equation $$ y'' =y$$ you can write it as the pair of first order differential equations $$ u' = y \\ y' = u$$ which is the vector diff-eq $$ \vec w' = A\vec w$$ where $\vec w=(u,y)^T$ and $$ A = \begin{pmatrix}0&1\\1&0\end{pmatrix}$$

Answer 2

Let $$ \vec{x} = \begin{pmatrix} x_1 \\ x_2 \\ \end{pmatrix} = \begin{pmatrix} y_1' \\ y_2' \\ \end{pmatrix} = \vec{y}'. $$ Then $$ \vec{x}' = \begin{pmatrix} x_1' \\ x_2' \\ \end{pmatrix} = \begin{pmatrix} a & b \\ c & d \\ \end{pmatrix} \begin{pmatrix} y_1\\ y_2 \\ \end{pmatrix} = \begin{pmatrix} a & b \\ c & d \\ \end{pmatrix} \vec{y}. $$ In coordinates, $$ \begin{pmatrix} \vec{x}' \\ \vec{y}' \\ \end{pmatrix} = \begin{pmatrix} x_1' \\ x_2' \\ y_1' \\ y_2' \\ \end{pmatrix} = \begin{pmatrix} 0 & 0 & a & b \\ 0 & 0 & c & d \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \\ y_1 \\ y_2 \\ \end{pmatrix} = \begin{pmatrix} \vec{x} \\ \vec{y} \\ \end{pmatrix}, $$ or $$ \begin{pmatrix} \vec{x}' \\ \vec{y}' \\ \end{pmatrix} = \begin{pmatrix} 0 & A \\ I & 0 \\ \end{pmatrix} \begin{pmatrix} \vec{x} \\ \vec{y} \\ \end{pmatrix}, $$ where $$ A= \begin{pmatrix} a & b \\ c & d \\ \end{pmatrix} \mbox{ and } I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \\ \end{pmatrix}. $$