Systems of First-Order Linear ODEs: Eigenvalue Method

A system of two first-order linear ODEs, x′ = ax + by and y′ = cx + dy, is written compactly as x′ = Ax with a 2×2 matrix A. The eigenvalue method assumes solutions of the form x(t) = v·eλt; substituting gives Av = λv, so λ must be an eigenvalue of A and v its eigenvector. Each eigenpair produces one exponential mode, and the general solution combines them.

This is the standard tool for anything coupled: two connected water tanks, predator and prey populations near equilibrium, two masses joined by springs, or two electrical loops sharing a component. The eigenvalues reveal the system's fate — growth, decay, or oscillation — before any formula is written down.

Key Concepts

• Characteristic equation: det(A − λI) = 0, a quadratic whose roots are the eigenvalues λ1, λ2.
• Eigenvector: a non-zero vector v with (A − λI)v = 0; it gives the direction of that exponential mode in the phase plane.
• General solution (distinct eigenvalues): x(t) = C1v1eλ1t + C2v2eλ2t.
• Complex eigenvalues α ± iβ: produce real solutions eαtcos(βt) and eαtsin(βt) via Euler's formula — oscillations with growing or decaying amplitude.

Formulas

• For A = [[a,b],[c,d]]: λ2 − (tr A)λ + det A = 0, where tr A = a + d and det A = ad − bc.
• Eigenvector for λ: solve (a−λ)v1 + b·v2 = 0 — pick any convenient non-zero solution.
• Complex case: from v·e(α+iβ)t take real and imaginary parts as two real solutions.
• Assumption: A is constant. Limit: if A is defective (repeated eigenvalue, one eigenvector), the pure exponential ansatz is incomplete without the t·eλt term.

Worked Example

Solve x′ = 4x − 2y, y′ = x + y. Here A = [[4,−2],[1,1]], tr A = 5, det A = 4·1 − (−2)·1 = 6. Characteristic equation: λ2 − 5λ + 6 = 0, so λ1 = 2, λ2 = 3. For λ = 2: (A − 2I)v = [[2,−2],[1,−1]]v = 0 gives v1 = v2, choose v1 = [1,1]. For λ = 3: (A − 3I)v = [[1,−2],[1,−2]]v = 0 gives v1 = 2v2, choose v2 = [2,1]. General solution: [x;y] = C1e2t[1;1] + C2e3t[2;1]. Check the first equation with C1 = C2 = 1: x = e2t + 2e3t, y = e2t + e3t; x′ = 2e2t + 6e3t while 4x − 2y = 4e2t + 8e3t − 2e2t − 2e3t = 2e2t + 6e3t. They match, so the solution is verified.

Common Misconceptions

• Eigenvectors are unique. Any non-zero multiple of an eigenvector is still an eigenvector — [1,1] and [2,2] describe the same mode.
• Complex eigenvalues mean there is no real solution. Wrong: the real and imaginary parts of the complex solution give two perfectly real oscillatory solutions.

Real-World Applications

• Coupled mass-spring systems: eigenvalues give the normal-mode frequencies of vibration.
• Compartment models in pharmacokinetics — drug moving between blood and tissue — are linear systems solved this way.
• Linearised predator-prey and epidemic models near equilibrium predict whether populations oscillate or settle.

The eigenvalue method turns x′ = Ax into the algebraic problem det(A − λI) = 0, with each eigenpair contributing a mode C·v·eλt. For 2×2 systems the characteristic equation is λ2 − (tr A)λ + det A = 0. It is the universal first step for coupled linear systems across physics, biology, and engineering.