Systems of ODEs | Mathematics - Wyatt's Notes

4.1 First-Order Linear Systems

A system of first-order linear ODEs can be written in matrix form:

$\mathbf{x}' = A\mathbf{x} + \mathbf{f}(t)$

Where $A$ is an $n \times n$ matrix and $\mathbf{x}, \mathbf{f} \in \mathbb{R}^n$.

4.2 Homogeneous Systems with Constant Coefficients

For $\mathbf{x}' = A\mathbf{x}$Try $\mathbf{x} = \mathbf{v}e^{\lambda t}$:

$\lambda \mathbf{v} = A\mathbf{v}$

So $\lambda$ is an eigenvalue of $A$ and $\mathbf{v}$ is the corresponding eigenvector.

Case 1: $A$ has $n$ distinct real eigenvalues. The general solution is

$\mathbf{x} = c_1 \mathbf{v}_1 e^{\lambda_1 t} + \cdots + c_n \mathbf{v}_n e^{\lambda_n t}$

Case 2: $A$ has a repeated eigenvalue $\lambda$ with algebraic multiplicity $m$ and geometric Multiplicity $k \lt m$. Include terms involving $t^j e^{\lambda t}$ where generalized Eigenvectors fill out the solution space.

Case 3: Complex eigenvalues $\lambda = \alpha \pm i\beta$ with eigenvector $\mathbf{v} = \mathbf{a} \pm i\mathbf{b}$. The real solutions are $e^{\alpha t}(\mathbf{a}\cos(\beta t) - \mathbf{b}\sin(\beta t))$ and $e^{\alpha t}(\mathbf{a}\sin(\beta t) + \mathbf{b}\cos(\beta t))$.

4.3 The Matrix Exponential

Definition. $e^{At} = \sum_{k=0}^{\infty} \frac{A^k t^k}{k!}$.

Theorem 4.1. The solution to $\mathbf{x}' = A\mathbf{x}$ with $\mathbf{x}(0) = \mathbf{x}_0$ is $\mathbf{x}(t) = e^{At}\mathbf{x}_0$.

Proposition 4.2. If $A$ is diagonalizable as $A = PDP^{-1}$ Then $e^{At} = Pe^{Dt}P^{-1}$ Where $e^{Dt} = \mathrm{diag}(e^{\lambda_1 t}, \ldots, e^{\lambda_n t})$.

4.4 Worked Example: Distinct Real Eigenvalues

Problem. Solve $\mathbf{x}' = \begin{pmatrix} 0 & 1 \\ -2 & -3 \end{pmatrix}\mathbf{x}$.

Solution. Characteristic equation: $\det(A - \lambda I) = \lambda^2 + 3\lambda + 2 = (\lambda + 1)(\lambda + 2) = 0$. Eigenvalues: $\lambda_1 = -1$, $\lambda_2 = -2$.

For $\lambda_1 = -1$: $(A + I)\mathbf{v} = \begin{pmatrix} 1 & 1 \\ -2 & -2 \end{pmatrix}\mathbf{v} = \mathbf{0}$. $\mathbf{v}_1 = \begin{pmatrix} 1 \\ -1 \end{pmatrix}$.

For $\lambda_2 = -2$: $(A + 2I)\mathbf{v} = \begin{pmatrix} 2 & 1 \\ -2 & -1 \end{pmatrix}\mathbf{v} = \mathbf{0}$. $\mathbf{v}_2 = \begin{pmatrix} 1 \\ -2 \end{pmatrix}$.

$\mathbf{x}(t) = c_1 \begin{pmatrix} 1 \\ -1 \end{pmatrix} e^{-t} + c_2 \begin{pmatrix} 1 \\ -2 \end{pmatrix} e^{-2t}$. $\blacksquare$

4.5 Worked Example: Complex Eigenvalues

Problem. Solve $\mathbf{x}' = \begin{pmatrix} 0 & -2 \\ 1 & 0 \end{pmatrix}\mathbf{x}$.

4.6 Worked Example: Repeated Eigenvalues

Problem. Solve $\mathbf{x}' = \begin{pmatrix} 2 & 1 \\ -1 & 4 \end{pmatrix}\mathbf{x}$.

4.7 Fundamental Matrix

Definition. A fundamental matrix $\Phi(t)$ for the system $\mathbf{x}' = A\mathbf{x}$ is an $n \times n$ matrix whose columns form a fundamental set of solutions.

Proposition 4.3. $\Phi(t)$ satisfies $\Phi' = A\Phi$ And the general solution is $\mathbf{x}(t) = \Phi(t)\mathbf{c}$ for $\mathbf{c} \in \mathbb{R}^n$.

Proposition 4.4. The matrix exponential $e^{At}$ is a fundamental matrix with $e^{A \cdot 0} = I$. Any fundamental matrix can be written as $\Phi(t) = e^{At}\Phi(0)$.

4.8 Matrix Exponential Properties

Theorem 4.5. The matrix exponential satisfies:

1. $e^{A \cdot 0} = I$
2. $\frac{d}{dt}e^{At} = Ae^{At} = e^{At}A$
3. $e^{At}e^{As} = e^{A(t+s)}$
4. $(e^{At})^{-1} = e^{-At}$
5. If $AB = BA$ Then $e^{A+B} = e^A e^B$

Proof of (1). $e^{A \cdot 0} = \sum_{k=0}^{\infty} \frac{A^k 0^k}{k!} = I$. $\blacksquare$

Proof of (2). $\frac{d}{dt}e^{At} = \sum_{k=1}^{\infty} \frac{A^k t^{k-1}}{(k-1)!} = A\sum_{j=0}^{\infty} \frac{A^j t^j}{j!} = Ae^{At}$. Since $A$ commutes with itself, $Ae^{At} = e^{At}A$. $\blacksquare$

Proof of (4). From (3) with $s = -t$: $e^{At}e^{-At} = e^{A(t-t)} = e^0 = I$. $\blacksquare$

4.9 Phase Portrait Analysis for 2D Systems

For the linear system $\mathbf{x}' = A\mathbf{x}$ with $A$ a $2 \times 2$ matrix, the qualitative Behaviour near the origin is determined by the eigenvalues:

Remark. The trace-determinant plane provides a convenient classification. Let $\tau = \mathrm{tr}(A)$ and $\Delta = \det(A)$. The eigenvalues satisfy $\lambda^2 - \tau\lambda + \Delta = 0$ So:

$\lambda = \frac{\tau \pm \sqrt{\tau^2 - 4\Delta}}{2}$

• $\tau^2 - 4\Delta > 0$: real eigenvalues (node or saddle)
• $\tau^2 - 4\Delta \lt 0$: complex eigenvalues (spiral or center)
• $\tau^2 - 4\Delta = 0$: repeated eigenvalues (proper or improper node)

Stability is determined by the sign of $\tau$: stable if $\tau \lt 0$Unstable if $\tau > 0$.

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The trace-determinant plane classifies 2D linear systems. The parabola $\tau^2 = 4\Delta$ separates real from complex eigenvalues; the $\tau = 0$ line separates stable from unstable. The x-axis represents the trace $\tau$ and the y-axis represents $\Delta$.

4.10 Nonhomogeneous Systems

For $\mathbf{x}' = A\mathbf{x} + \mathbf{f}(t)$If $\Phi(t)$ is a fundamental matrix for the Homogeneous system, the general solution is

$\mathbf{x}(t) = \Phi(t)\mathbf{c} + \Phi(t)\int \Phi^{-1}(s)\mathbf{f}(s)\, ds$

Worked Example. Solve $\mathbf{x}' = \begin{pmatrix} 1 & 0 \\ 0 & 2 \end{pmatrix}\mathbf{x} + \begin{pmatrix} e^t \\ 0 \end{pmatrix}$.

4.11 Common Mistakes

flowchart TD
    A[4_Systems Of Odesx] --> B[Key Concepts]
    A --> C[Core Principles]
    A --> D[Practical Applications]
    B --> E[Fundamental definitions]
    C --> F[Design patterns]
    D --> G[Real-world usage]

Intuition

Systems of ODEs describe how multiple quantities evolve together, each potentially influencing the others. A linear system $\mathbf{x}' = A\mathbf{x}$ decomposes along the eigenvectors of $A$: each eigenvector defines a natural direction of motion, and its eigenvalue determines whether the solution grows, decays, or oscillates along that direction. The matrix exponential $e^{At}$ is the fundamental object — it maps initial conditions to solutions at time $t$. When eigenvalues are complex, the solution spirals; when real and negative, it decays toward the origin; when real and positive, it explodes. The trace-determinant plane classifies all 2D linear systems into nodes, saddles, spirals, and centres.

4.11 Common Mistakes

Mistake 1: Forgetting to account for all initial conditions When solving a system of $n$ ODEs, the general solution must contain $n$ arbitrary constants. Students often omit constants or assume the initial conditions are automatically satisfied. Always apply all initial conditions to determine every constant, and verify the solution satisfies each equation in the system.

Mistake 2: Using the wrong eigenvalue-eigenvector pairs When a matrix has repeated eigenvalues, the eigenvectors may not span the full space, requiring generalised eigenvectors. A common error is writing the same eigenvector for both occurrences of a repeated eigenvalue instead of constructing the Jordan chain. Check the geometric multiplicity against the algebraic multiplicity.

Mistake 3: Confusing the matrix exponential $e^{At}$ with $e^A e^{t}$ The identity $e^{At} = e^{At}$ holds trivially, but $e^{A+B} = e^A e^B$ only when $AB = BA$. Students sometimes write $e^{At} = (e^A)^t$ or attempt to exponentiate component-wise. For diagonalisable matrices, use $e^{At} = Pe^{Dt}P^{-1}$ where $D$ is the diagonal matrix of eigenvalues.

Cross-References

• Stability and Phase Plane Analysis: The phase portrait classification of 2D systems uses the eigenvalue analysis developed for matrix systems.
• Second-Order Linear ODEs: Systems of first-order ODEs can be derived from a single second-order ODE by introducing new variables.
• Laplace Transforms: The Laplace transform converts systems of ODEs into algebraic equations in the frequency domain.
• Partial Derivatives: The Jacobian matrix and matrix exponential are fundamental tools from linear algebra used throughout systems theory.