Systems Summary

Differential Equations · Axiom Academy

Recapping Unit 5: turning higher-order equations into vector systems, solving by eigenvalues, and reading the phase plane. Any higher-order linear equation converts to a first-order vector system by introducing state variables — this unlocks linear algebra as a solution tool. The eigenvalues of A control everything: real parts set growth/decay, imaginary parts set oscillation, and the three eigenvalue cases (real-distinct, complex, repeated) each build the general solution differently. Solution vectors are linearly independent for all time exactly when their Wronskian is nonzero — and Abel's formula shows a constant-coefficient system's Wronskian is never zero unless it starts at zero. The trace-determinant plane packages every critical-point classification (node, saddle, spiral, center) into one map, built purely from and . Nonlinear systems are classified the same way after linearizing at each equilibrium: compute the Jacobian there, then read its trace and determinant. Core Concept From Scalar to Vector Introducing state variables turns any single higher-order equation into a first-order system. The new coefficient matrix A (the companion matrix) carries the same information as the original equation, but now the machinery of linear algebra — eigenvalues, eigenvectors — applies directly. When to use: the very first step whenever a problem is handed to you as a higher-order scalar DE.

This is the written version of the interactive lesson above. See the full Differential Equations course.