<s>[INST] $y''=y$, $x''=-x$. Write this equation in terms of the first-order system I'm having trouble with my homework on higher-order equations and their equivalent systems. :(
This is the problem:
Write this equation in terms of the first-order system.
$$\left\{ \begin{array}{l} \frac{d^2x}{dt^2}=-x \\ \frac{d^2y}{dt^2}=y \end{array} \right.$$
(Hint: Write each second-order equation as two first-order equations.
 [/INST] If $X= \left( \begin{array}{c} x' \\ x \end{array} \right)$, then $X'= \left( \begin{array}{c} x'' \\ x' \end{array} \right) = \left( \begin{array}{c} -x \\ x' \end{array} \right)= \left( \begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array} \right) \left( \begin{array}{c} x' \\ x \end{array} \right)=AX$. You can do the same thing for $y$.
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