\chapter{Linear Systems of Differential Equations}
Although we have a good idea on how to solve many single linear differential equations, it's often useful to think about linear systems of equations, like
\begin{equation*}
	\begin{cases}
		x_1' = 3tx_1 - x_2 + t^2 \\
		x_2' = -x_1 + e^tx_2 - e^{2t}
	\end{cases}
	\to
	\begin{bmatrix}
		x_1 \\ 
		x_2
	\end{bmatrix}' = \begin{bmatrix}
		3t & -1 \\
		-1 & e^t
	\end{bmatrix} \begin{bmatrix}
		x_1 \\
		x_2
	\end{bmatrix} + \begin{bmatrix}
		t^2 \\
		-e^{2t}
	\end{bmatrix}.
\end{equation*}
This matrix form, $\vec{x'} = A\vec{x} + \vec{f}$, where $A$ is a square matrix, is called normal form.\\

% Solutions to systems
\input{./linearSystems/solutions/solutions.tex}
% Homogenous
\input{./linearSystems/homogeneousSystems/homogeneousSystems.tex}
% Non-Homogenous
\input{./linearSystems/heterogeneousSystems/heterogeneousSystems.tex}
% Higher-order linear diffeq's as systems of 1st order linear diffeq's
\input{./linearSystems/systemsAndHigherOrder/systemsAndHigherOrder.tex}