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So consider the following linear non-autonomous ODE: \begin{equation} \begin{bmatrix} \dot{x}_1 \\ \dot{x}_2 \end{bmatrix} = \begin{bmatrix} t^2 & 1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \; \; \; \;(1) \end{equation} where for convenience we have denoted $x_1$ for $x_1(t)$ and $x_2$ for $x_2(t)$ with $t \in \mathbb{R}$.

I am curious if there exists initial conditions (apart from the trivial one) such that the solution remains bounded as $t \rightarrow \infty$. This seems like a pretty standard problem, does anybody have an idea how to go about with this?

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$\begin{bmatrix}\dot{x}_1\\\dot{x}_2\end{bmatrix}=\begin{bmatrix}t^2&1\\1&0\end{bmatrix}\begin{bmatrix}x_1\\x_2\end{bmatrix}$

$\begin{cases}\dot{x}_1=t^2x_1+x_2\\\dot{x}_2=x_1\end{cases}$

$\therefore\ddot{x}_1=t^2\dot{x}_1+2tx_1+\dot{x}_2=t^2\dot{x}_1+2tx_1+x_1$

$\ddot{x}_1-t^2\dot{x}_1-(2t+1)x_1=0$

But since this ODE unfortunately relates to Triconfluent Heun equation (http://dlmf.nist.gov/31.12#E4 and http://www.maplesoft.com/support/help/Maple/view.aspx?path=HeunT), it is very difficult to analyse further.

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