Systems of ODEs

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 $\tau > 0$.

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