Systems of Linear ODEs

Focus on Linear Time-Invariant Systems

This video sets the stage for the types of differential equations we will study in this section.

Form of the Solution

When solving linear differential equations, it is often a good strategy to seek a solution of a particular form. In this video, we show the particular form and then see how this leads to an eigenvalue problem.

Eigenvalue Problem

In the next two videos, we solve the eigenvalue problem that comes from finding solutions to systems of linear differential equations. The first illustrates what a solution to the eigenvalue problem looks like.

The second video steps you through the process of calculating the eigenvalues and eigenvectors by hand.

General Solution

Now we use the solution to the eigenvalue to write the general solution to the differential equations.

Coordinate Mapping and Phase Portrait

Next, we’re going to define a new set of states through a coordinate transformation. This will help us construct the phase portrait.

Another Example

We work through another example of solving a system of linear differential equations. This time we calculate eigenvalues and eigenvectors with computer software. And we construct the phase portrait without the aid of a coordinate transformation.

What if the Eigenvalues are Complex?

Here are two more videos where we use a somewhat different transformation to illustrate the nature of the solutions when the eigenvalues are complex.