Systems of Differential Equations - Theory & Concepts

Systems of Differential Equations

$Learning Objectives$

Solve linear systems of differential equations using the elimination method.
Solve homogeneous linear systems using the matrix eigenvalue method.
Construct and interpret phase portraits for 2D linear systems.
Classify the stability and type of critical points based on eigenvalues.

A system of differential equations consists of two or more equations involving derivatives of two or more unknown functions. These systems often model interacting populations (Predator-Prey), coupled oscillators, or chemical reactions.

$Linear System Form$

A linear system of first-order differential equations expresses the derivatives of multiple dependent variables as linear combinations of the variables themselves.

Linear System Form (2D)

A 2x2 system of linear first-order differential equations.

\frac{dx}{dt} = ax + by

\frac{dy}{dt} = cx + dy

Variables

Symbol	Description	Unit
$x$	First dependent variable	-
$y$	Second dependent variable	-
$t$	Independent variable (e.g., time)	-
$a, b, c, d$	Constant coefficients	-

Matrix Notation of a Linear System

The compact matrix representation of a linear system.

\mathbf{x}' = A\mathbf{x}

Variables

Symbol	Description	Unit
$\mathbf{x}$	Vector of dependent variables	-
$\mathbf{x}'$	Derivative of the state vector with respect to time	-
$A$	Coefficient matrix	-

Method 1: Elimination

This method involves eliminating variables systematically until you are left with a single higher-order differential equation for one variable. It is analogous to algebraic substitution/elimination.

$Elimination Steps$

Rewrite: Use the differential operator notation ( $D = d/dt$ ). Write equations so variables and their derivatives are on one side.
- $x' = 2x - y \implies (D-2)x + y = 0$
- $y' = x + 2y \implies -x + (D-2)y = 0$
Operate: Treat the operators $D$ like algebraic polynomials. Multiply entire equations by appropriate operators to match terms (like finding a common denominator).
Eliminate: Add or subtract equations to eliminate one dependent variable (e.g., eliminate $x$ ).
Solve: You will now have a single higher-order homogeneous DE in terms of the remaining variable ( $y$ ). Solve it using the auxiliary equation method to find $y(t)$ .
Back-Substitute: Substitute $y(t)$ back into an original first-order equation (if possible) or a derived equation to find $x(t)$ without introducing extra arbitrary constants.

Method 2: Matrix Method (Eigenvalues)

For a linear, homogeneous system with constant coefficients represented as $\mathbf{x}' = A\mathbf{x}$ , the solution is intimately tied to the eigenvalues and eigenvectors of matrix $A$ .

$Eigenvalue Procedure$

Assume a solution of the form $\mathbf{x} = \mathbf{v}e^{\lambda t}$ . Substituting into the system gives $A\mathbf{v} = \lambda\mathbf{v}$ .

Find Eigenvalues ( $\lambda$ ): Solve the characteristic equation $\det(A - \lambda I) = 0$ for $\lambda$ . This will be a polynomial of degree $n$ for an $n \times n$ system.
Find Eigenvectors ( $\mathbf{v}$ ): For each eigenvalue $\lambda_i$ , solve the homogeneous system $(A - \lambda_i I)\mathbf{v}_i = \mathbf{0}$ to find the corresponding eigenvector.
Form Solution: The general solution is a linear combination of the fundamental solutions. The form depends on the type of eigenvalues:
- Distinct Real Eigenvalues ( $\lambda_1 \neq \lambda_2$ ): The two eigenvectors are independent and give two exponential modes.
- Repeated Real Eigenvalues ( $\lambda_1 = \lambda_2 = \lambda$ ): Check the eigenspace dimension. Two independent eigenvectors give two ordinary modes; a defective 2×2 matrix requires a generalized eigenvector satisfying $(A-\lambda I)\mathbf{w}=\mathbf{v}$ .
- Complex Conjugate Eigenvalues ( $\lambda = \alpha \pm \beta i$ ): Solutions involve $e^{\alpha t}(\cos(\beta t)\mathbf{u} - \sin(\beta t)\mathbf{w})$ and $e^{\alpha t}(\sin(\beta t)\mathbf{u} + \cos(\beta t)\mathbf{w})$ where $\mathbf{v} = \mathbf{u} + i\mathbf{w}$ is the complex eigenvector.

General Solution for Distinct Real Eigenvalues

The solution for a system where all eigenvalues are real and distinct.

\mathbf{x}(t) = c_1 e^{\lambda_1 t}\mathbf{v}_1 + c_2 e^{\lambda_2 t}\mathbf{v}_2 + \dots

Variables

Symbol	Description	Unit
$\mathbf{x}(t)$	State vector over time	-
$c_1, c_2$	Arbitrary constants determined by initial conditions	-
$\lambda_1, \lambda_2$	Distinct eigenvalues	-
$\mathbf{v}_1, \mathbf{v}_2$	Corresponding eigenvectors	-
$t$	Time	s

Phase Portrait Visualizer

Visualize the behavior of a 2x2 linear system by plotting its Phase Portrait. The trajectory $(x(t), y(t))$ traces a curve in the $xy$ -plane over time. The origin $(0,0)$ is the critical point.

$Interactive Simulation$

Interact with the phase portrait simulation below to visualize 2D linear systems and critical points.

Phase Portrait

\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} -1.0 & -1.0 \\ 1.0 & -1.0 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}

Critical Point (0,0): Stable Spiral

a_{11}

-1.0

a_{12}

-1.0

a_{21}

1.0

a_{22}

-1.0

\text{Tr}(A) = -2.00

\det(A) = 2.00

\Delta = \text{Tr}^2 - 4\det = -4.00

x range: [-5, 5]

$Classification of Critical Points (2D Linear Systems)$

The qualitative behavior of the system $\mathbf{x}' = A\mathbf{x}$ near the origin is determined by the eigenvalues of matrix $A$ and, for repeated real eigenvalues, by the dimension of the eigenspace.

Eigenvalues	Type of Critical Point	Stability
Real, distinct, both positive ( $\lambda_1 > \lambda_2 > 0$ )	Proper node (two independent eigendirections)	Unstable (Source)
Real, distinct, both negative ( $\lambda_1 < \lambda_2 < 0$ )	Proper node (two independent eigendirections)	Asymptotically Stable (Sink)
Real, distinct, opposite signs ( $\lambda_1 < 0 < \lambda_2$ )	Saddle Point	Unstable
Real, repeated, positive ( $\lambda_1 = \lambda_2 > 0$ )	Node: proper with two eigenvectors; improper if defective	Unstable (Source)
Real, repeated, negative ( $\lambda_1 = \lambda_2 < 0$ )	Node: proper with two eigenvectors; improper if defective	Asymptotically Stable (Sink)
Complex conjugate with positive real part ( $\alpha \pm i\beta, \alpha > 0$ )	Spiral Point	Unstable (Source)
Complex conjugate with negative real part ( $\alpha \pm i\beta, \alpha < 0$ )	Spiral Point	Asymptotically Stable (Sink)
Pure imaginary ( $\pm i\beta, \alpha = 0$ )	Center	Stable (but not asymptotically)

Matrix Eigenvalue Solution Workflow

Matrix Eigenvalue Solution for a 2×2 Linear System

Solve a homogeneous constant-coefficient system by eigenmodes, branching when a repeated eigenvalue has too few independent eigenvectors.

This workflow applies to a homogeneous linear system x′ = Ax with a constant 2×2 matrix A on the interval. Form det(A − λI) = 0, find each eigenspace, and determine whether the eigenvectors span the state space. Real distinct eigenvalues supply independent exponential modes. A complex-conjugate pair for a real matrix supplies a real basis from the real and imaginary parts of a complex mode. For a repeated eigenvalue with two independent eigenvectors, use two ordinary exponential modes. For a defective 2×2 repeated eigenvalue, find w satisfying (A − λI)w = v and include e^(λt)(tv + w). Larger systems may require longer Jordan chains and are outside this chart. Apply any supplied initial data, substitute the result into the original system, and check the initial state.

Visual Relationships in Linear Systems

Component Equations Become One State-Vector Equation

Collect dependent variables into a state vector and place their constant coefficients in matching rows of A.

Write x′ and y′ as a column vector

Place each equation's coefficients in one row of A

Express the system as x′ = Ax

Use the same state ordering in A, eigenvectors, and initial data

Elimination and Matrix Methods Reduce Different Structures

Both methods solve the same constant-coefficient linear system, but elimination produces a scalar higher-order equation while the matrix method works with eigenmodes.

Elimination

Apply differential operators
Solve a higher-order scalar DE
Back-substitute for remaining variables

Matrix eigenvalue method

Find eigenvalues and eigenvectors of A
Build vector-valued exponential modes
Apply initial data to the state vector

Repeated Eigenvalues Depend on Eigenspace Dimension

A repeated algebraic eigenvalue does not by itself determine the solution form; count independent eigenvectors before adding a generalized mode.

Two independent eigenvectors | two ordinary eʳᵗv modes

One eigenvector for a repeated 2×2 root | solve (A − λI)w = v

Defective case | add eʳᵗ(tv + w)

Verify the complete solution in the original system

Eigenvalue Signs Predict the Linear Phase Portrait

For a real 2×2 linear system without zero eigenvalues, eigenvalue signs or real parts distinguish nodes, saddles, spirals, and centers; zero eigenvalues need separate analysis.

Real eigenvalues with opposite signs | saddle

Real eigenvalues with both signs positive | unstable node

Real eigenvalues with both signs negative | asymptotically stable node

Complex pair α ± iβ | spiral if α ≠ 0; center if α = 0

Key Takeaways

Systems of DEs relate multiple dependent variables.
Elimination Method: Transforms the system into a single higher-order DE using differential operators $D$ .
Matrix Method: Solves $\mathbf{x}' = A\mathbf{x}$ using eigenvalues and eigenvectors. Check the eigenspace dimension for repeated roots; a defective repeated root needs a generalized eigenvector mode.
Phase Portraits: A graphical representation of solutions $(x(t), y(t))$ plotted parametrically in the $xy$ -plane.
Critical Point Classification: For a hyperbolic equilibrium (no eigenvalue has zero real part), eigenvalue real parts determine linear stability; zero-real-part cases need further analysis.

$References$

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