Say I have a mass $m_1$ attached to a spring $s_1$, the other end of which is fixed. Then, I have another mass $m_2$ attached to a spring $s_2$, the other end of which is attached to $m_1$. Gravity applies to the masses.
(ceiling) - $s_1$ - $m_1$ - $s_2$ - $m_2$
The spring constants for $s_1$ and $s_2$ are $k_1$ and $k_2$, respectively.
I need someone to look over my equilibrium equations because I've done so a few times and they look correct, but my solution doesn't look correct.
$$ \begin{aligned} m_1y_1'' - k_1y_1 - k_2y_1 + k_2y_2 &= 0\\ m_2y_2'' - k_2y_2 + k_2y_1 &= 0 \end{aligned} $$
Rearrange a little:
$$ \begin{aligned} m_1y_1'' - (k_1 + k_2)y_1 + k_2y_2 &= 0\\ m_2y_2'' + k_2y_1 - k_2y_2 &= 0\\\\ m_1y_1'' &= (k_1 + k_2)y_1 - k_2y_2\\ m_2y_2'' &= - k_2y_1 + k_2y_2\\\\ y_1'' &= \frac{k_1 + k_2}{m_1}y_1 - \frac{k_2}{m_1}y_2\\ y_2'' &= \frac{-k_2}{m_2}y_1 + \frac{k_2}{m_2}y_2\\ \end{aligned} $$
The constant values are $k_1=3$, $k_2=2$, $m_1=1$, $m_2=1$:
$$ \begin{aligned} y_1'' &= 5y_1 - 2y_2\\ y_2'' &= -2y_1 + 2y_2\\ \end{aligned} $$
In matrix form I have:
$$ \bar y'' = \bar A \bar y, \bar A=\begin{bmatrix}5 & -2\\\ -2 & 2\end{bmatrix} $$
If I assume a solution $y$ has the form $\bar y = \bar v e^{\omega t}$, then
$$ \omega^2 \bar v e^{\omega t} = \bar A \bar v e^{\omega t}\\ \omega^2 \bar v = \bar A \bar v $$
Sub. $\omega^2 = \lambda$ gives me $$ \lambda \bar v = \bar A \bar v $$
From this point, I'm solving for eigenvalues and eigenvectors in the usual way. I get
$$ \lambda_1 = 1, \lambda_2 = 6. $$
With $\lambda_1$, I get $\bar v = \begin{bmatrix}1 \\\ 2\end{bmatrix}$. With $\lambda_2$, I get $\bar v = \begin{bmatrix}-2 \\\ 1\end{bmatrix}$.
So I get the solution
$$ \bar y = c_1 \begin{bmatrix}-2 \\\ 1\end{bmatrix}e^{\sqrt 6 t} + c_2 \begin{bmatrix}1 \\\ 2\end{bmatrix}e^{\sqrt t} $$
Since the square roots really give me two solutions each, I can manipulate this into
$$ \bar y = c_1 \begin{bmatrix}-2 \\\ 1\end{bmatrix}cosh(\sqrt 6 t) + c_2 \begin{bmatrix}1 \\\ 2\end{bmatrix} cosh(t) $$
I can't find an error, but the solution doesn't make sense to me; I'd expect the system to oscillate and the solution doesn't.
What's my mistake?