System of higher order Differential Equations

Question

Sorry asking out of interest but is it possible for there to be a system of ODEs like

$X'' + X' = AX$

if so could one give an example of system which could be represented as this

Answer 1

If $\mathbf A = (a_{ij})_{ij} $ and if $\mathbf X = [x_1,\dots,x_n]^{\text{T}}$, we can write your system as

$$ \begin{cases} x_1'' + x_1 ' = a_{11}x_1 + \dots + a_{1n}x_n \\ x_2'' + x_2 ' = a_{21}x_1 + \dots + a_{2n}x_n \\ \vdots \\ x_n'' + x_n ' = a_{n1}x_1 + \dots + a_{nn}x_n \\ \end{cases} $$

For example, if $n=2$ and $\mathbf A = \operatorname{Id}$, we get

$$ \begin{cases} x_1''+x_1' = x_1\\ x_2'' +x_2' = x_2 \end{cases} $$

which has solution

$$ \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} c_1 \operatorname{e}^{-\frac{1}{2}t(1+\sqrt 5)} +c_2 \operatorname{e}^{-\frac{1}{2}t(1-\sqrt 5)} \\ c_3 \operatorname{e}^{-\frac{1}{2}t(1+\sqrt 5)} +c_4 \operatorname{e}^{-\frac{1}{2}t(1-\sqrt 5)} \\ \end{bmatrix} $$

Answer 2

Calling $X = X_1, \dot X_1 = X_2$ we have

$$ \left(\begin{array}{c}\dot X_1\\ \dot X_2\end{array}\right)=\left(\begin{array}{cc} 0 & I\\ A & -I\end{array}\right)\left(\begin{array}{c}X_1\\ X_2\end{array}\right) $$