homogeneous linear systems with constant coefficients solutions of systems of odes

Homogeneous Linear Systems with Constant

CoefficientsSolutions of Systems of ODEs

Important Linear AlgebraRecall Eigenvalues and

Eigenvectors

And Linear Independencean

dAre linearly

independent ifdet

Linear Systems of Ordinary Differential

Equations

Let’s rewrite this in matrix form:

Or

Linear Systems of Ordinary Differential

Equations

What if

was an eigenvector of ?

Then:

Linear Systems of Ordinary Differential

Equations

What if

was an eigenvector of ?

Then:

Linear Systems of Ordinary Differential

Equations

What if

was an eigenvector of ?

Then:

Or:

Linear Systems of Ordinary Differential

Equations

What if

was an eigenvector of ?

Then:

Or:

Linear Systems of Ordinary Differential

Equations

What if

was an eigenvector of ?

Then:

Or:

Or:

Linear Systems of Ordinary Differential

Equations

What if

was an eigenvector of ?

But these are two independent,

separable equations!

Linear Systems of Ordinary Differential

Equations

What if

was an eigenvector of ?

Solutions

Linear Systems of Ordinary Differential

Equations

What if

was an eigenvector of ?

Solution

But if is an

Eigenvectoris an

Eigenvector

Two Specific Solutions

For a 2x2 System with Eigenvalues and Eigenvectorsan

dand

Specific Solutions:

or

Example

Find Two Solutions to The Set of Linear Differential

Equations

Linear Combinations of SolutionsRemember, For Linear

Equations, if

is a solution and

is a solution, then

is also a solution.

Linear Combinations of Solutions

How can you tell?

Rememberand

Linear Combinations of Solutions

How can you tell?

Rememberandso

Linear Combinations of Solutions

How can you tell?

Linear Combinations of Solutions

How can you tell?

Because

Are Scalars

Linear Combinations of Solutions

How can you tell?

Because

Are Scalars

Linear Combinations of Solutions

How can you tell?

So

Fundamental Set of Solutions

Additionally, if the Eigenvectors are linearly independent

Then

Form a Fundamental Set of Solutions,

is the general solution.

det

and

Why?Consider the Wronskian

det

Never 0

Only 0 if

are linearly dependent

det

So To Solve

Determine All Eigenvalues and Eigenvectors of

General Solution takes the form

Plug in 0 and use initial conditions to find

Summary

• Eigenvalues and Eigenvectors Can Be Used To Find General Solution of Systems of Equations

• If Eigenvectors are linearly independent, then we can find general solutions.

Questions?