Systems of Differential Equations - Theory & Concepts

Systems of Differential Equations

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

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

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

Or in matrix notation: $\mathbf{x}' = A\mathbf{x}$

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 ( $\lambda_1 \neq \lambda_2$ ):
  
  $\mathbf{x}(t) = c_1 e^{\lambda_1 t}\mathbf{v}_1 + c_2 e^{\lambda_2 t}\mathbf{v}_2$
- Repeated Real Eigenvalues ( $\lambda_1 = \lambda_2 = \lambda$ ): Usually involves a term with $te^{\lambda t}$ and finding a generalized eigenvector.
- 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.

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.

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 entirely determined by the eigenvalues ( $\lambda_1, \lambda_2$ ) of matrix $A$ .

Eigenvalues	Type of Critical Point	Stability
Real, distinct, both positive ( $\lambda_1 > \lambda_2 > 0$ )	Improper Node	Unstable (Source)
Real, distinct, both negative ( $\lambda_1 < \lambda_2 < 0$ )	Improper Node	Asymptotically Stable (Sink)
Real, distinct, opposite signs ( $\lambda_1 < 0 < \lambda_2$ )	Saddle Point	Unstable
Real, repeated, positive ( $\lambda_1 = \lambda_2 > 0$ )	Proper / Degenerate Node	Unstable (Source)
Real, repeated, negative ( $\lambda_1 = \lambda_2 < 0$ )	Proper / Degenerate Node	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)

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 ( $\lambda$ ) and eigenvectors ( $\mathbf{v}$ ). The general form is $\mathbf{x}(t) = c_1 e^{\lambda_1 t}\mathbf{v}_1 + c_2 e^{\lambda_2 t}\mathbf{v}_2$ .
Phase Portraits: A graphical representation of solutions $(x(t), y(t))$ plotted parametrically in the $xy$ -plane.
Critical Point Classification: The origin's stability is completely determined by the eigenvalues (e.g., negative real parts mean a sink/stable; opposite signs mean a saddle/unstable).

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