# 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.