homogeneous linear systems with constant coefficients solutions of systems of odes
TRANSCRIPT
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?