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Differential Equations Unit 8: Systems of first-order DEs

Systems of first-order equations model several interacting quantities. Linear systems are solved with eigenvalues and eigenvectors.

Covers: Matrix methods, eigenvalue solutions, phase portraits

Study guide

A system of first-order linear equations can be written in matrix form x′ = Ax, where x(t) is a vector of unknown functions. Higher-order equations can be converted into systems by introducing new variables, like x₁ = y and x₂ = y′.

The eigenvalue method finds solutions of the form x = e^(λt)v, where λ is an eigenvalue of A and v is a corresponding eigenvector. With n independent eigenvectors, the general solution is a combination: x = c₁e^(λ₁t)v₁ + c₂e^(λ₂t)v₂ + …. Complex eigenvalues α ± βi give real solutions involving e^(αt)cos βt and e^(αt)sin βt. A repeated eigenvalue with only one eigenvector needs a generalized eigenvector w satisfying (A − λI)w = v, giving solutions like te^(λt)v + e^(λt)w.

Phase portraits plot solution trajectories in the x₁x₂-plane. For 2×2 systems, the eigenvalues classify the origin (an equilibrium point). Real eigenvalues of the same sign give a node: stable if negative (trajectories approach), unstable if positive. Real eigenvalues of opposite signs give a saddle, which is unstable. Complex eigenvalues give spirals, stable if the real part is negative and unstable if positive, and purely imaginary eigenvalues give centers (closed orbits). The origin is asymptotically stable exactly when every eigenvalue has negative real part. Eigenvector directions show the straight-line trajectories.

A fundamental matrix Φ(t) has independent solutions as its columns, and the general solution is x = Φ(t)c. The matrix exponential is e^(tA) = Φ(t)Φ(0)⁻¹, and the solution with x(0) = x₀ is x(t) = e^(tA)x₀. For a 2×2 matrix with trace T and determinant D, the eigenvalues solve λ² − Tλ + D = 0. D 0 and T² − 4D > 0 it is a node, if T² − 4D 0 means unstable.

A repeated eigenvalue with only one independent eigenvector gives a degenerate (improper) node, where trajectories become tangent to the single eigenvector direction. A repeated eigenvalue with two independent eigenvectors gives a star node, where every trajectory is a straight line through the origin. For a nonlinear system x′ = f(x, y), y′ = g(x, y), equilibria are the points where f = g = 0. Near each one, the system behaves like the linear system whose matrix is the Jacobian [f_x f_y; g_x g_y] evaluated there. If that linearization is a saddle, node or spiral, the nonlinear equilibrium is the same type, but a linear center isn't conclusive.

Key ideas

• Write the system as x′ = Ax.
• Each eigenpair (λ, v) gives a solution e^(λt)v.
• Real distinct eigenvalues: nodes or saddles. Complex: spirals or centers.
• Phase portraits show trajectories in the plane.
• Stability: all eigenvalues with negative real part → stable.

Formulas and rules

Common mistakes

• Forgetting that complex eigenvalues give real solutions with sines and cosines.
• Calling a center stable in the asymptotic sense.

Words to know

Matrix form of a linear system

x′ = Ax

Solution from an eigenpair (λ, v)

e^(λt)v

Saddle (phase plane)

Real eigenvalues of opposite sign; unstable.

Stable spiral

Complex eigenvalues with negative real part.

Asymptotically stable origin

Every eigenvalue has negative real part.

Matrix form of a system

x′ = Ax

Eigenvalue method

Solutions e^(λt)v from eigenpairs of A.

Generalized eigenvector

w with (A − λI)w = v for a repeated eigenvalue.

Phase portrait

Solution trajectories plotted in the x₁x₂-plane.

Node

Real eigenvalues of the same sign; stable if negative.

Center

Purely imaginary eigenvalues; closed orbits.

Unstable spiral

Complex eigenvalues with positive real part.

Sample practice questions

19 more questions like these, from level 1 to 5, are in the practice set.