Incomplete proof for an analytic solution for [[../02 Areas/Physics/Damped Harmonic Oscillators|Damped Harmonic Oscillators]] that is just *wrong*. We can encode the state of this system with a [[State Space Representation]] by encoding each point in some 2 [[Dimension]] [[../Math/Linear Subspace|Linear Space]] with each coordinate representing $(x, x)$. $\huge y =\mat{x\\x''} $ We can then evaluate: $ \begin{align} \let y &= \mat{ x\\x'} \\ y' &= \mat{x'\\x''} \\ y' &= \mat{ \alpha^{-1}\pa{-x''-kx }\\ -\alpha x' -kx } \\ &= \mat{ -\alpha^{-1} x'' - kx \\ -\alpha x' -kx } \\ &= \mat{ -\alpha^{-1} x'' - kx \\ -\alpha x' -kx } \\ &= \mat{ -\alpha^{-1} \pa{-\alpha x'-kx} - kx \\ -\alpha x' -kx } \\ &= \mat{ x' + (\alpha^{-1}-k)x \\ -\alpha x' - kx }\\ &=\mat{ (\alpha^{-1}-k)x + x' \\ - kx -\alpha x' }\\ &=\mat{ \alpha^{-1}-k & 1\\ -k & -\alpha }\mat{x\\x'} \end{align} $ Which gives us: $y' =\mat{ \alpha^{-1}-k & 1\\ -k & -\alpha }y $ $ \begin{align} y &=A_{0}+e^{ \mat{ \alpha^{-1}-k & 1\\ -k & -\alpha }}\\ y' &= \end{align} $