Linear ODE Systems

Definition

A system of homogeneous linear first-order ODEs with constant coefficients:

$${\dot{x}}_1=a{x}_1+b{x}_2,{\dot{x}}_2=c{x}_1+d{x}_2,$$

written in matrix form as $\dot{\mathbf{x}}=\mathbf{Ax}$.

Intuition

The matrix $\mathbf{A}$ stretches and rotates the state vector $\mathbf{x}$; the natural directions of stretching are the eigenvectors. Along each eigendirection the system decouples into a scalar ODE $\dot{v}=\lambda v$, so the solution is a sum of exponentials weighted by eigenvalues — a direct vector-space generalisation of the scalar characteristic root cases.

Formal Description

The exponential ansatz $\mathbf{x}(t)=\mathbf{v}{e}^{\lambda t}$ converts the system into the eigenvalue problem

$$\mathbf{Av}=\lambda \mathbf{v}.$$

The characteristic equation for the $2\times 2$ matrix is

$$\det (\mathbf{A}-\lambda \mathbf{I})={\lambda }^2-(a+d)\lambda +(ad-bc)=0.$$

Each eigenvalue ${\lambda }_k$ with eigenvector ${\mathbf{v}}_k$ gives a solution ${\mathbf{v}}_k{e}^{{\lambda }_{k}t}$.

Key Results

Three cases arise, analogous to the scalar characteristic root cases:

Eigenvalues	General solution
Distinct real ${\lambda }_1\ne {\lambda }_2$	$\mathbf{x}=c_1{\mathbf{v}}_1{e}^{{\lambda }_{1}t}+c_2{\mathbf{v}}_2{e}^{{\lambda }_{2}t}$
Complex conjugates $\lambda =\alpha \pm i\beta$	Separate real and imaginary parts using Euler’s formula
Repeated eigenvalue	Second solution involves $t{e}^{\lambda t}$ (as in scalar case)

The principle of superposition applies since the system is linear.

Applications

• Phase-plane analysis and stability of equilibria in two-dimensional dynamical systems.
• Coupled mechanical or electrical oscillators.
• Reducing any $n$th-order scalar ODE to a first-order system (companion matrix form) for numerical or analytical treatment.

Trade-offs

• Finding eigenvectors requires solving $(\mathbf{A}-\lambda \mathbf{I})\mathbf{v}=0$; repeated eigenvalues may yield defective matrices requiring generalised eigenvectors.
• For complex eigenvalues, Euler’s formula is needed to extract real-valued solution components.
• Scales well to $n\times n$ systems in principle, but eigendecomposition becomes expensive for large $n$.

Links

• Characteristic Roots
• Homogeneous Second-Order ODE
• Principle of Superposition