Linear systems and matrix methods are crucial tools for analyzing systems of differential equations. They provide a compact way to represent and solve complex problems in various fields, from biology to engineering.

By using matrices, we can efficiently solve and analyze multiple equations simultaneously. This approach allows us to understand the behavior of interconnected variables over time, predict system stability, and identify critical points in multi-dimensional spaces.

Matrix Notation for Linear Systems

Representing Systems with Matrices

• Matrix notation provides a compact and efficient way to represent systems of linear differential equations facilitating easier manipulation and analysis
• General form of a system of linear differential equations in matrix notation $\frac{dx}{dt} = Ax + b(t)$
  • $\frac{dx}{dt}$ represents the vector of derivatives of the dependent variables
  • A denotes the coefficient matrix containing coefficients of dependent variables arranged in rows and columns
  • x(t) represents the dependent variables as functions of time typically arranged as a column vector
  • b(t) contains non-homogeneous terms or forcing functions arranged as a column vector
• For homogeneous systems b(t) = 0 simplifying the equation to $\frac{dx}{dt} = Ax$
• Order of the system determined by number of first-order differential equations equal to number of rows (and columns) in coefficient matrix A

Examples and Applications

• Two-dimensional predator-prey system: $\frac{dx}{dt} = ax - bxy$ $\frac{dy}{dt} = -cy + dxy$ Represented in matrix form as: $\begin{bmatrix} \frac{dx}{dt} \\ \frac{dy}{dt} \end{bmatrix} = \begin{bmatrix} a & -b \\ d & -c \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}$
• Three-dimensional system modeling chemical reactions: $\frac{dA}{dt} = -k_1A + k_2B$ $\frac{dB}{dt} = k_1A - (k_2 + k_3)B + k_4C$ $\frac{dC}{dt} = k_3B - k_4C$ Represented in matrix form as: $\begin{bmatrix} \frac{dA}{dt} \\ \frac{dB}{dt} \\ \frac{dC}{dt} \end{bmatrix} = \begin{bmatrix} -k_1 & k_2 & 0 \\ k_1 & -(k_2 + k_3) & k_4 \\ 0 & k_3 & -k_4 \end{bmatrix} \begin{bmatrix} A \\ B \\ C \end{bmatrix}$

Solving Linear Systems with Matrices

Eigenvalue Method for Homogeneous Systems

• Eigenvalue method fundamental technique for solving homogeneous systems of linear differential equations
• Find eigenvalues λ of coefficient matrix A using characteristic equation $det(A - λI) = 0$
  • I represents the identity matrix of the same size as A
• Determine eigenvectors v corresponding to each eigenvalue by solving $(A - λI)v = 0$
• General solution for system with distinct real eigenvalues linear combination of terms $ce^{λt}v$
  • c denotes an arbitrary constant
• For complex eigenvalues a ± bi solution involves terms $e^{at}(c_1\cos(bt) + c_2\sin(bt))v$
  • v represents real or imaginary part of complex eigenvector
• Repeated eigenvalues may require generalized eigenvectors leading to solutions with terms $te^{λt}v$

Non-homogeneous Systems and Variation of Parameters

• Variation of parameters method used to solve non-homogeneous systems
• Involves fundamental matrix solution of homogeneous system and integration of its inverse with non-homogeneous term
• Steps for variation of parameters:
  1. Find general solution of corresponding homogeneous system
  2. Construct fundamental matrix Φ(t) using linearly independent solutions
  3. Compute inverse of fundamental matrix Φ⁻¹(t)
  4. Integrate $\int Φ^{-1}(t)b(t)dt$ to find particular solution
  5. Combine homogeneous and particular solutions for complete solution

General and Particular Solutions for Linear Systems

Constructing General Solutions

• General solution of homogeneous system linear combination of n linearly independent solutions
  • n represents order of system
• For non-homogeneous systems general solution sum of:
  1. General solution of corresponding homogeneous system
  2. Particular solution of non-homogeneous system
• Wronskian determinant verifies linear independence of solution set and constructs fundamental matrix solution
• Matrix exponential $e^{At}$ provides compact representation of fundamental matrix solution for homogeneous systems
  • Used to express general solution

Finding Particular Solutions

• Method of undetermined coefficients finds particular solutions for systems with specific non-homogeneous terms (polynomials, exponentials, trigonometric functions)
• Steps for method of undetermined coefficients:
  1. Identify form of non-homogeneous term
  2. Propose particular solution with unknown coefficients
  3. Substitute proposed solution into original system
  4. Equate coefficients to solve for unknown parameters
• Laplace transform method effective alternative for systems with constant coefficients
  • Transforms differential equations into algebraic equations
  • Solve resulting system of algebraic equations
  • Apply inverse Laplace transform to obtain solution in time domain

Analyzing Solutions of Linear Systems

Stability and Long-term Behavior

• Eigenvalues of coefficient matrix determine stability and long-term behavior of homogeneous system solutions
• Stability criteria:
  • Negative real parts of all eigenvalues indicate asymptotic stability
  • Positive real parts for any eigenvalue result in instability
  • Zero real parts lead to neutral stability or potential instability
• Phase plane visualizes behavior of two-dimensional systems
  • Trajectories represent solutions over time
• Critical points (equilibrium solutions) found by solving:
  • $Ax = 0$ for homogeneous systems
  • $Ax + b = 0$ for non-homogeneous systems with constant b

Classification of Critical Points and System Behavior

• Critical point classification determined by eigenvalues and eigenvectors of coefficient matrix:
  • Nodes (stable or unstable) real eigenvalues with same sign
  • Saddles real eigenvalues with opposite signs
  • Spirals (stable or unstable) complex conjugate eigenvalues with non-zero real parts
  • Centers purely imaginary eigenvalues
• Limit cycles closed trajectories in phase plane
  • Other nearby trajectories approach or diverge from limit cycles
  • Indicate periodic behavior in system
• Linear systems exhibit various behaviors depending on eigenvalues and eigenvectors:
  • Growth exponential increase in magnitude
  • Decay exponential decrease in magnitude
  • Oscillation periodic variation
  • Combinations simultaneous growth/decay and oscillation