 # How can I write the following equation as a first-order vector differential equation? <mtable c hughy46u 2022-05-29 Answered
How can I write the following equation as a first-order vector differential equation?
$\begin{array}{r}m\frac{{\mathrm{d}}^{2}x}{{\mathrm{d}}^{2}t}+2\gamma m\frac{\mathrm{d}x}{\mathrm{d}t}+kx=0.\end{array}$
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Introduce a new variable $y\left(t\right)=\stackrel{˙}{x}\left(t\right)$. Then
$\begin{array}{rl}\stackrel{˙}{x}\left(t\right)& =\phantom{\rule{4.8em}{0ex}}y\left(t\right)\\ \stackrel{˙}{y}\left(t\right)& =-\frac{k}{m}x\left(t\right)-2\gamma y\left(t\right)\end{array}$
Then if we take $v=\left[\begin{array}{c}x\\ y\end{array}\right]$ and $A=\left[\begin{array}{cc}0& 1\\ -\frac{k}{m}& -2\gamma \end{array}\right]$, we have the following vector differential equation:
$\frac{\mathrm{d}}{\mathrm{d}t}v\left(t\right)=Av\left(t\right).$

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