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Expert ∞ Differential Equations, Abstract Algebra & Applied Math Topic 22 of 97

Systems of Linear ODEs

Solve with eigenvalues and eigenvectors.

Key idea

Solve with eigenvalues and eigenvectors.

In depth

x′ = Ax is solved using the eigenvalues and eigenvectors of A: x(t) = Σcᵢe^(λᵢt)vᵢ. Complex eigenvalues give oscillation, and the real part sets growth or decay. Matrix exponential e^(At) gives the general solution.

Worked example

A = [1 2; 2 1]: λ = 3, −1; x = c₁e^(3t)(1, 1) + c₂e^(−t)(1, −1).

Formula

x′ = Ax

Real-world uses

Common mistake

Using eigenvectors with the wrong eigenvalue.

How to learn this

  1. Read the key idea and say it in your own words.
  2. Work through the worked example, then try a similar problem.
  3. Use the interactive lesson’s visual and mini-quiz to check the pattern.

Frequently asked questions

What is Systems of Linear ODEs?

Solve with eigenvalues and eigenvectors. It is a topic at the expert level in Differential Equations, Abstract Algebra & Applied Math.

What is the formula for Systems of Linear ODEs?

The key formula is x′ = Ax.

Where is Systems of Linear ODEs used in real life?

Real-world examples: Modelling two linked tanks, coupled pendulums or interacting species together.

What should I learn before Systems of Linear ODEs?

Make sure you are comfortable with Convolution and Impulse Response, the previous topic in Differential Equations, Abstract Algebra & Applied Math.

What is a common mistake with Systems of Linear ODEs?

Using eigenvectors with the wrong eigenvalue.

How can I practise Systems of Linear ODEs?

Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.

What comes after Systems of Linear ODEs?

Move on to Phase Plane Analysis next.

Related expert topics in Differential Equations, Abstract Algebra & Applied Math