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Consider the system of higher order differential equations

$\begin{array}{rcl} y^{\,\prime\prime} & = & t^{-1} y^{\,\prime} + 3 y - t z + (\sin t) z^{\,\prime} + e^{4 t}, \\ z^{\,\prime\prime} & = & y - 7 z^{\,\prime}. \end{array}$
Rewrite the given system of two second order differential equations as a system of four first order linear differential equations of the form $\boldsymbol{\vec{y}^{\,\prime}} = P(t) \boldsymbol{\vec{y}} + \boldsymbol{\vec{g}}(t)$ . Use the following change of variables

$\boldsymbol{\vec{y}}(t) = \left\lbrack \begin{array}{r} y_1(t) \\ y_2(t) \\ y_3(t) \\ y_4(t) \end{array} \right\rbrack = \left\lbrack \begin{array}{r} y(t) \\ y^{\,\prime}(t) \\ z(t) \\ z^{\,\prime}(t) \end{array} \right\rbrack.$

$\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\!\strut\\\!\strut\\\end{array}}\right[$	$y_1^{\,\prime}$	$\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\!\strut\\\!\strut\\\end{array}}\right]$
	$y_2^{\,\prime}$
	$y_3^{\,\prime}$
	$y_4^{\,\prime}$

=

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

$\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\!\strut\\\!\strut\\\end{array}}\right[$	$y_1$	$\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\!\strut\\\!\strut\\\end{array}}\right]$
	$y_2$
	$y_3$
	$y_4$

+

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