---
title: "Systems of Differential Equations and Eigenvalue Solutions"
description: "Several coupled unknowns become one vector equation. The eigenvalues of the coefficient matrix give the growth rates and the eigenvectors give the directions along which the system decouples."
canonical: https://lightmysky.com/learn/mathematics/systems-of-differential-equations-and-eigenvalue-solutions-mt_BbOqJ4S1ur
source: https://lightmysky.com/learn/mathematics/systems-of-differential-equations-and-eigenvalue-solutions-mt_BbOqJ4S1ur.md
retrieved: 2026-09-12
---

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# Systems of Differential Equations and Eigenvalue Solutions

Several coupled unknowns become one vector equation. The eigenvalues of the coefficient matrix give the growth rates and the eigenvectors give the directions along which the system decouples.

Subject: Mathematics · Area: Differential Equations · Ages 21 to 22
Page: https://lightmysky.com/learn/mathematics/systems-of-differential-equations-and-eigenvalue-solutions-mt_BbOqJ4S1ur

## Ready when they can

- Write a coupled pair as a single matrix equation
- Solve a two-by-two system through eigenvalues and eigenvectors
- Convert a second-order equation into a first-order system

## Lesson: One matrix, two linked unknowns

Sharks and sardines in one sea, two water tanks joined by a pipe, your arms and the swing they push: when two quantities each feed the other's change, one equation cannot hold them. Two linked unknowns become one vector equation, x prime equals A times x. The matrix A holds all four rates in one place, and from here the whole system moves as a single object.

**Example.** A diagonal matrix with two and three on its diagonal stretches one axis by two and the other by three. Its eigenvalues sit on the diagonal: two and three. Each axis direction is an eigenvector, since the matrix only stretches it. The vector one one is no eigenvector here: multiplying returns two three, which points another way.

For a general two by two system, subtract lambda on the diagonal and set the determinant to zero. That characteristic equation gives the growth rates, and each rate's hidden line gives its direction. To convert, name velocity v as y prime. For y double prime plus four y prime plus three y, the second line reads v prime equals minus three y minus four v.

**Tip.** Test any claimed eigenvector by multiplying: the result must run parallel to the claim. One one fails for the diagonal two three matrix, since it returns two three, which points elsewhere.

**Recap.** Pack the pair into a matrix, read growth from eigenvalues, direction from eigenvectors, and convert orders with a velocity variable.

## Practice

13 questions on this page, each with its working shown.

## Needs first

- [Solving Initial Value Problems with Laplace Transforms](https://lightmysky.com/learn/mathematics/solving-initial-value-problems-with-laplace-transforms-mt_0BhjPVNUwl)
- [Eigenvalues and Eigenvectors](https://lightmysky.com/learn/mathematics/eigenvalues-and-eigenvectors-mt_TVqqaw11qa)
- [Diagonalisation and Powers of a Matrix](https://lightmysky.com/learn/mathematics/diagonalisation-and-powers-of-a-matrix-mt_w15uCVhdhX)

## Opens up

- [Phase Portraits, Equilibria and Stability](https://lightmysky.com/learn/mathematics/phase-portraits-equilibria-and-stability-mt_ULuZk4lmGr)