$\textsf{6.1.1 Show that}\\$
$$y=\begin{bmatrix}
c_1e^{2x} + c_2e^{3x}\\
2c_1e^{2x} + c_2e^{3x}
\end{bmatrix}\\$$
$\textsf{is a solution of} \\$
$$Y'=\left[\begin{array}{rr}
4 & -1 \\2 & 1
\end{array}\right]Y\\$$
$\textsf{6.1.2 Show that}\\$
$$Y'=\left[\begin{array}{rrrr}
c_1e^{2x}& + c_2e^{3x}& -\frac{x}{6}&-\frac{11}{36} \\ \\
2c_1e^{2x}& + c_2e^{3x}& -\frac{2x}{6}&-\frac{1}{18}
\end{array}\right]\\$$
$\textsf{is a solution of} \\$
$$Y'=\left[\begin{array}{rr}
4 & -1 \\2 & 1
\end{array}\right]Y
+\left[\begin{array}{rr}
1 \\x
\end{array}\right]$$
ok I realize these are simple but still the examples given confused me.
I assume the first step is to span
$$y=\begin{bmatrix}
c_1e^{2x} + c_2e^{3x}\\
2c_1e^{2x} + c_2e^{3x}
\end{bmatrix}\\$$
$\textsf{is a solution of} \\$
$$Y'=\left[\begin{array}{rr}
4 & -1 \\2 & 1
\end{array}\right]Y\\$$
$\textsf{6.1.2 Show that}\\$
$$Y'=\left[\begin{array}{rrrr}
c_1e^{2x}& + c_2e^{3x}& -\frac{x}{6}&-\frac{11}{36} \\ \\
2c_1e^{2x}& + c_2e^{3x}& -\frac{2x}{6}&-\frac{1}{18}
\end{array}\right]\\$$
$\textsf{is a solution of} \\$
$$Y'=\left[\begin{array}{rr}
4 & -1 \\2 & 1
\end{array}\right]Y
+\left[\begin{array}{rr}
1 \\x
\end{array}\right]$$
ok I realize these are simple but still the examples given confused me.
I assume the first step is to span