Mr. Integral
Math notes/Differential Equations/Systems and Phase Planes

Linear Systems and Eigenmodes

2 MIN READ · PRACTISE AT YOUR OWN PACE   ·   FREE

LESSON 1 OF 5See the unit map ↗

Convert higher-order equations to first-order systems and use eigenvectors to solve them.

Builds on Transfer Functions, Poles and Response · Eigenvalues and Eigenspaces

The bigger question: How do coupled variables move together?

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The idea

A state vector collects enough quantities to predict future evolution. For a second-order scalar equation, displacement and velocity form a two-component state. A constant linear system has the form x′=Ax\mathbf x'=A\mathbf x; its eigenvectors identify directions preserved by the dynamics.

Visual guide

VISUAL GUIDEA fast mode fades before the slow mode
For A = [[−2, 1], [1, −2]] and initial state (1, 0), the two components are (e⁻ᵗ ± e⁻³ᵗ)/2. Their difference decays faster than their common part, so the state approaches the slow eigenvector direction (1, 1).0010.27520.5530.82541.1tstate
  • x₁(t)
  • x₂(t)
For A = [[−2, 1], [1, −2]] and initial state (1, 0), the two components are (e⁻ᵗ ± e⁻³ᵗ)/2. Their difference decays faster than their common part, so the state approaches the slow eigenvector direction (1, 1).

Method and assumptions

If Av=λvA\mathbf v=\lambda\mathbf v, then eλtve^{\lambda t}\mathbf v solves the system. With a full basis of eigenvectors, combine these modes and solve for coefficients from the initial state. If no eigenbasis exists, generalized eigenvectors or a matrix exponential are needed.

Worked example: coupled decay

For A=(−211−2)A=\begin{pmatrix}-2&1\\1&-2\end{pmatrix}, vectors (1,1)(1,1) and (1,−1)(1,-1) have eigenvalues −1,−3-1,-3. Initial state (1,0)(1,0) decomposes as half of each, giving x=12e−t(1,1)+12e−3t(1,−1)\mathbf x=\tfrac12e^{-t}(1,1)+\tfrac12e^{-3t}(1,-1).

PAUSE & THINKA quick check, not a grade

Try it yourself.

Choose an answer and explain your reasoning to yourself. Use a hint if you get stuck.

If Av=λv, which function solves x′=Ax along that mode?

Hint 1 · Find a starting point

Differentiate the proposed mode and use the eigenvector equation.

Hint 2 · Take the next step

Both derivative and A times the state must give λeλte^{\lambda t}v.

Show the reasoning

Answer: x(t)=eλte^{\lambda t}v

An eigenvector evolves by the scalar exponential eλte^{\lambda t}, so x′=Ax.

Before moving on: what would make one of the other answers wrong? Saying why is part of understanding.

Worked example: a scalar equation as a system

For y′′+3y′+2y=0y''+3y'+2y=0, choose x1=y,x2=y′x_1=y,x_2=y'. Then x′=(01−2−3)x\mathbf x'=\begin{pmatrix}0&1\\-2&-3\end{pmatrix}\mathbf x. Its eigenvalues −1,−2-1,-2 agree with the scalar characteristic roots.

Interpreting the result

Eigenvectors describe state directions, which may mix quantities with different physical units. Define the state and scaling before interpreting a phase-plane angle as a physical angle.

Practice

  1. Write a system for y′′+4y=0y''+4y=0.
  2. Solve x1′=−x1,x2′=−2x2x_1'=-x_1,x_2'=-2x_2 from (2,3)(2,3).
  3. Which mode dominates that solution at late times?
Show worked solutions
  1. x1′=x2,x2′=−4x1x_1'=x_2,x_2'=-4x_1.
  2. (2e−t,3e−2t)(2e^{-t},3e^{-2t}).
  3. The e−te^{-t} mode decays more slowly and dominates when its coefficient is nonzero.

Further study

MIT OpenCourseWare: Differential Equations provides a full university course with additional lectures and exercises.

MAKE IT YOURS

Pause before the next idea.

Can you explain how the visualization connects to this lesson’s goal? If a step still feels uncertain, put this lesson on your review list and try the check again another day.

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