$$y=\begin{bmatrix}

c_1e^{2x} + c_2e^{3x}\\

2c_1e^{2x} + c_2e^{3x}

\end{bmatrix}\\$$

$\textsf{is a solution of} \\$

$$Y'=\left[\begin{array}{rr}

4 & -1 \\2 & 1

\end{array}\right]Y\\$$

$\textsf{6.1.2 Show that}\\$

$$Y'=\left[\begin{array}{rrrr}

c_1e^{2x}& + c_2e^{3x}& -\frac{x}{6}&-\frac{11}{36} \\ \\

2c_1e^{2x}& + c_2e^{3x}& -\frac{2x}{6}&-\frac{1}{18}

\end{array}\right]\\$$

$\textsf{is a solution of} \\$

$$Y'=\left[\begin{array}{rr}

4 & -1 \\2 & 1

\end{array}\right]Y

+\left[\begin{array}{rr}

1 \\x

\end{array}\right]$$

ok I realize these are simple but still the examples given confused me.

I assume the first step is to span