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I'm looking at a system x'=Ax as follows: $$\left[ \begin{matrix}x'(t) \\y'(t)\end{matrix} \right] = \left[\begin{matrix}0 & 1\\4/t^2 & -1/t\end{matrix}\right]\left[ \begin{matrix}x(t) \\y(t)\end{matrix} \right]$$ and I want to find the homogeneous solution and therefore $\Phi(t)$, the fundamental matrix.

My problem is that, whilst I know several ways to solve this type of system, I do not know how to deal with the $t$ in matrix A. The first method I considered was finding the eigenvalues and eigenvectors, but I end up with characteristic polynomial $\lambda^2 - \frac{\lambda}{t}-\frac{4}{t^2}$, and thus eigenvalues $\lambda=\frac{1/t \pm \sqrt{1/t^2+16/t^2}}{2}=\frac{1 \pm \sqrt{17}}{2t}$ and I figure the eigenvectors will get messy, let alone the fundamental matrix. I considered finding the matrix exponential $e^{At}$, but this seemed messy from the start.

What is the best way to solve a system like this?

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Eigenvectors and eigenvalues are not helpful here. You don't want a matrix exponential when the system is not autonomous. You could use a "time-ordered exponential", but that's unnecessarily complicated here.

Note that if you plug in $y = x'$ to the equation for $y'$, you get $ t^2 x'' + t x' - 4 x = 0$, which is an Euler differential equation.

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