Prerequisites
Matrix form and solution structure
Many systems can be written compactly with vectors and matrices.
Linear system
A general linear system is
where:
$\mathbf{x}(t)$ is the unknown vector
$A(t)$ is a coefficient matrix
$\mathbf{g}(t)$ is a forcing vector
If $A$ is constant and $\mathbf{g}(t)=\mathbf{0}$, the system is linear autonomous homogeneous:
This is the most important case because it is fully analyzable with linear algebra.
Superposition
For a homogeneous linear system:
if $\mathbf{x}_1$ and $\mathbf{x}_2$ are solutions, then $\mathbf{x}_1 + \mathbf{x}_2$ is also a solution
if $c$ is a constant, then $c\mathbf{x}_1$ is also a solution
That means the set of solutions forms a vector space.
Dimension of the solution space
For an $n \times n$ first-order linear homogeneous system, the general solution depends on $n$ constants. Equivalently, you need $n$ independent initial conditions to determine a unique solution.
Linear autonomous systems
The canonical form is
with constant matrix $A$.
Why this case matters
The behavior of the system is controlled by the matrix $A$:
eigenvalues determine growth, decay, oscillation, and stability
eigenvectors determine invariant directions
the matrix exponential gives the exact solution
For a $2 \times 2$ system,
or equivalently,
Decoupling idea
If you can find a change of variables that diagonalizes or simplifies $A$, the system may split into independent scalar ODEs. This is the linear-algebra viewpoint behind most solution methods.
Eigenmodes as an ODE application
The eigenvalue definitions, characteristic-equation derivation, eigenspaces, and repeated-root cases are developed in the Eigenvalues note. Here we use only their consequence for time-dependent solutions: if $A\mathbf{v}=\lambda\mathbf{v}$, then
is an eigenmode of $\mathbf{x}'=A\mathbf{x}$. The mode interpretation is:
| Eigenvalue pattern | ODE behavior |
|---|---|
| Real $\lambda<0$ | Exponential decay |
| Real $\lambda>0$ | Exponential growth |
| $\lambda=0$ | Neutral mode |
| $\alpha\pm\beta i$ | Oscillation with envelope $e^{\alpha t}$ |
| Repeated/defective eigenvalue | Generalized-eigenvector terms such as $t e^{\lambda t}$ may occur |
If enough independent eigenvectors exist, the general homogeneous solution is a linear combination of these modes. The table is the application recap; see Eigenvalues for how to compute and classify the eigenpairs.
The matrix exponential
The exact solution of
is
where $e^{At}$ is the matrix exponential.
Definition
This is analogous to the scalar exponential series.
Diagonalizable case
Using the diagonalization result from the Eigenvalues note, with $D = \operatorname{diag}(\lambda_1,\dots,\lambda_n)$, the matrix exponential is
and
This is the cleanest computational route; the construction and validity of $P$ and $D$ belong to the Eigenvalues note.
Practical use
You often do not compute the full series. Instead, use:
diagonalization
eigenvector decomposition
Jordan form when necessary
software or a calculator for large systems
Nonhomogeneous linear systems
The forced system
combines natural dynamics from $A$ with external input $\mathbf{g}(t)$.
General solution structure
The solution is
where:
$\mathbf{x}_h$ solves the homogeneous system $\mathbf{x}' = A\mathbf{x}$
$\mathbf{x}_p$ is any particular solution of the full system
Variation of parameters
If $\Phi(t)$ is a fundamental matrix for $\mathbf{x}' = A\mathbf{x}$, then a particular solution can be written as
This is the matrix version of variation of parameters for scalar ODEs.
Common forcing types
Forcing terms often include:
constants
polynomials
exponentials
sines and cosines
combinations of the above
For constant-coefficient systems, undetermined coefficients can work when the forcing has a simple form.
Phase portraits and stability
For a two-dimensional autonomous system, the phase plane is the $(x,y)$ plane of states. A phase portrait shows representative trajectories.
Equilibria
An equilibrium point satisfies
For a linear system $\mathbf{x}' = A\mathbf{x}$, the only equilibrium is usually the origin unless the system is degenerate.
Stability ideas
An equilibrium is:
stable if nearby trajectories stay nearby
asymptotically stable if nearby trajectories converge to it
unstable if arbitrarily small perturbations can move away
For linear systems, the eigenvalues of $A$ determine this.
Classification for 2x2 systems
Let the eigenvalues of $A$ be $\lambda_1,\lambda_2$.
| Eigenvalue pattern | Typical behavior |
|---|---|
| Both real, negative | Stable node |
| Both real, positive | Unstable node |
| Opposite signs | Saddle, unstable |
| Complex with negative real part | Stable spiral |
| Complex with positive real part | Unstable spiral |
| Purely imaginary | Center in the ideal linear case |
If one eigenvalue is zero or repeated, examine the eigenvectors and the full normal form carefully.
Trace-determinant test for 2x2 systems
For
define:
Then the characteristic polynomial is
and the discriminant is
This quickly tells you whether the eigenvalues are real or complex and whether the origin is stable.
Nonlinear systems and linearization
A nonlinear system has the form
or more generally
Nonlinear systems can have multiple equilibria, limit cycles, bifurcations, and behavior that is not captured by linear theory alone.
Linearization
Near an equilibrium point $\mathbf{x}^*$, approximate the system by its Jacobian matrix:
where
This produces a linear system that approximates the nonlinear system near the equilibrium.
Interpreting the linearization
If the linearized system has eigenvalues with negative real part, the equilibrium is often locally asymptotically stable.
If it has an eigenvalue with positive real part, the equilibrium is unstable.
If eigenvalues lie on the imaginary axis or are zero, linearization may be inconclusive.
Caveat
Linearization is a local tool. It tells you about behavior near an equilibrium, not the full global dynamics.
Common modeling patterns
Coupled growth and decay
If one variable feeds another, the system may look like
with off-diagonal terms representing interaction rates.
Mass-spring systems
Second-order equations often become first-order systems by introducing velocity.
For
set
Then
This is a standard way to convert a higher-order ODE into a system.
Predator-prey models
Typical nonlinear interactions are modeled by
where growth of one species depends on the other. Equilibria are found by solving $x'=0$ and $y'=0$ simultaneously.
Electrical circuits
In circuits with capacitors and inductors, Kirchhoff's laws often produce systems for charge and current. These are frequently linear with constant coefficients.
Problem-solving workflow
When solving a system of ODEs, use a disciplined sequence.
Identify the type
Decide whether the system is:
linear or nonlinear
autonomous or nonautonomous
homogeneous or forced
one-dimensional, two-dimensional, or higher
Rewrite in matrix form
Put the system into
if possible.
Find equilibria
Set all derivatives to zero and solve the algebraic system.
Use the eigenvalue results
For linear systems, obtain eigenvalues and eigenvectors using the Eigenvalues note, then interpret the resulting modes in time.
Build the solution basis
Use:
eigenvectors if $A$ is diagonalizable
generalized eigenvectors if not
variation of parameters or undetermined coefficients for forcing
Apply initial conditions
Substitute the initial state to determine the constants.
Interpret the result
Check whether the solution:
grows or decays
oscillates
approaches equilibrium
matches the physical context
Formula sheet
Homogeneous linear system
Exact solution
Matrix exponential
Eigenmode interpretation
If $A\mathbf{v}=\lambda\mathbf{v}$, then $\mathbf{x}(t)=e^{\lambda t}\mathbf{v}$; see the eigenmode recap and the Eigenvalues note for computation.
Linearization near equilibrium
Forced linear system
Variation of parameters
Key pitfalls
Mixing up vector solutions with scalar solutions
Forgetting that one first-order system of size $n$ needs $n$ constants
Using scalar intuition when eigenvectors are not enough
Ignoring complex eigenvalues and their oscillatory interpretation
Treating linearization as a global solution method
Forgetting to apply initial conditions after finding the general solution
Quick intuition
Eigenvalues control time behavior.
Eigenvectors control directions.
The matrix exponential packages the whole solution.
Nonlinear systems are often best understood locally by linearization.
Sources
Stewart, Calculus: Early Transcendentals
Lay, Linear Algebra and Its Applications
Rosen, Discrete Mathematics and Its Applications
Boyce and DiPrima, Elementary Differential Equations and Boundary Value Problems
Blitzstein and Hwang, Introduction to Probability