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Suppose

$\begin{array}{rcl} y_1' & = & t^{2} y_1 + 3 y_2 + \sec\!\left(t\right), \\ y_2' & = & \sin\!\left(t\right) y_1 + t y_2 - 4. \end{array}$
This system of linear differential equations can be put in the form $\mathbf{y}' = P(t) \mathbf{y} + \mathbf{g}(t)$ . Determine $P(t)$ and $\mathbf{g}(t)$ .

$P(t) =$

$\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\end{array}}\right[$			$\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\end{array}}\right]$

$\mathbf{g}(t) =$

$\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\end{array}}\right[$		$\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\end{array}}\right]$