Section Linear System Basics
After exploring uncoupled, partially coupled, and fully coupled systems, we now turn to the most tractable and widely used class: linear systems with constant coefficients.
Their algebraic form is simple enough for exact analysis, yet flexible enough to model everything from harmonic oscillators to interacting populations. Our first task is to define what βlinearβ means and show how a short piece of matrix notation can replace a page-full of equations.
Subsection What Makes a System Linear?
A system is linear if every equation is linear with respect to its dependent variables (all to the power of one). For two variables \(x(t)\) and \(y(t)\) general form is
\begin{align*}
\frac{dx}{dt} \amp = a x + b y\\
\frac{dy}{dt} \amp = c x + d y
\end{align*}
where \(a\text{,}\) \(b\text{,}\) \(c\text{,}\) and \(d\) are constants. The system is therefore autonomous (no explicit \(t\)) and planar (two-dimensional).
Checkpoint 287.
Subsection The Matrix Formulation
Writing the same system once more,
\begin{align*}
\frac{dx}{dt} \amp = a x + b y\\
\frac{dy}{dt} \amp = c x + d y
\end{align*}
define the coefficient matrix and state vector
\begin{equation*}
A = \begin{bmatrix} a \amp b \\ c \amp d \end{bmatrix},
\qquad
Y = \begin{bmatrix} x \\ y \end{bmatrix},
\qquad
\frac{dY}{dt} = \begin{bmatrix} \dfrac{dx}{dt} \\ \dfrac{dy}{dt} \end{bmatrix}
\end{equation*}
Matrix multiplication then packages both equations into the single vector equation
\begin{equation*}
\frac{dY}{dt} = AY.
\end{equation*}
This compact format presents the system as a single equation, linking directly to linear algebra tools like eigenvalues, which we will explore later.
Subsection From Two to \(n\) Dimensions
\begin{equation*}
\frac{dY}{dt} = AY,
\qquad
Y = \begin{bmatrix} y_1 \\ \vdots \\ y_n \end{bmatrix},
\quad
A = \begin{bmatrix}
a_{11} \amp a_{12} \amp \dots \amp a_{1n} \\
\vdots \amp \amp \ddots \amp \vdots \\
a_{n1} \amp a_{n2} \amp \dots \amp a_{nn}
\end{bmatrix}.
\end{equation*}
The dimension of the system equals the length of \(Y\text{.}\) Our focus this chapter remains on the planar case (\(n=2\)) where geometry and algebra meet most clearly.
The number of dependent variables is called the dimension of the system. For example, the three-dimensional system:
Checkpoint 288.
Subsection π€ Wrap-Up
ποΈ \(\textbf{Key Takeaways...}\)
-
A system is linear if its unknowns appear without products, powers, or nonlinear functions.
-
Weβll focus on constant-coefficient linear systems, where the coefficients donβt change over time.
-
These systems can be written in vector form: \(\dfrac{dY}{dt} = AY\text{.}\)
Check Your Understanding.
Checkpoint 289. π€π Linear System Basics.
(a) πβ Linear or Not?
Decide whether each system is linear or nonlinear.
- \(\ds\quad\left\{\ \begin{array}{l} x' = 2x + 3y,\\y' = -x + y\end{array}\right.\)
- \(\ds\quad\left\{\ \begin{array}{l} x' = xy, \\ y' = x - y^2\end{array}\right.\)
- The product \(xy\) and the square \(y^2\) make this nonlinear.
- \(\ds\quad\left\{\ \begin{array}{l} x' = -4x + y, \\ y' = 5y\end{array}\right.\)
- \(\ds\quad\left\{\ \begin{array}{l} x' = e^x + y, \\ y' = -y\end{array}\right.\)
- The \(e^x\) term makes this nonlinear.
(b) πβ Identify Each System.
Select the linear systems.
- \(\ds\quad\left\{\ \begin{array}{l} x' = 4x - y,\\ y' = 2x + 3y\end{array}\right.\)
- \(\ds\quad\left\{\ \begin{array}{l} x' = xy,\\ y' = x - y^2\end{array}\right.\)
- \(\ds\quad\left\{\ \begin{array}{l} x' = -y,\\ y' = x\end{array}\right.\)
- \(\ds\quad\left\{\ \begin{array}{l} x' = \sin x + y,\\ y' = x\end{array}\right.\)
(c) πβ Benefits.
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