homogeneous linear systems with constant coefficients solutions of systems of odes
TRANSCRIPT
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Homogeneous Linear Systems with Constant
CoefficientsSolutions of Systems of ODEs
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Important Linear AlgebraRecall Eigenvalues and
Eigenvectors
And Linear Independencean
dAre linearly
independent ifdet
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Linear Systems of Ordinary Differential
Equations
Let’s rewrite this in matrix form:
Or
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Linear Systems of Ordinary Differential
Equations
What if
was an eigenvector of ?
Then:
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Linear Systems of Ordinary Differential
Equations
What if
was an eigenvector of ?
Then:
![Page 6: Homogeneous Linear Systems with Constant Coefficients Solutions of Systems of ODEs](https://reader034.vdocuments.us/reader034/viewer/2022052510/56649f3b5503460f94c5a2c2/html5/thumbnails/6.jpg)
Linear Systems of Ordinary Differential
Equations
What if
was an eigenvector of ?
Then:
Or:
![Page 7: Homogeneous Linear Systems with Constant Coefficients Solutions of Systems of ODEs](https://reader034.vdocuments.us/reader034/viewer/2022052510/56649f3b5503460f94c5a2c2/html5/thumbnails/7.jpg)
Linear Systems of Ordinary Differential
Equations
What if
was an eigenvector of ?
Then:
Or:
![Page 8: Homogeneous Linear Systems with Constant Coefficients Solutions of Systems of ODEs](https://reader034.vdocuments.us/reader034/viewer/2022052510/56649f3b5503460f94c5a2c2/html5/thumbnails/8.jpg)
Linear Systems of Ordinary Differential
Equations
What if
was an eigenvector of ?
Then:
Or:
Or:
![Page 9: Homogeneous Linear Systems with Constant Coefficients Solutions of Systems of ODEs](https://reader034.vdocuments.us/reader034/viewer/2022052510/56649f3b5503460f94c5a2c2/html5/thumbnails/9.jpg)
Linear Systems of Ordinary Differential
Equations
What if
was an eigenvector of ?
But these are two independent,
separable equations!
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Linear Systems of Ordinary Differential
Equations
What if
was an eigenvector of ?
Solutions
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Linear Systems of Ordinary Differential
Equations
What if
was an eigenvector of ?
Solution
But if is an
Eigenvectoris an
Eigenvector
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Two Specific Solutions
For a 2x2 System with Eigenvalues and Eigenvectorsan
dand
Specific Solutions:
or
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Example
Find Two Solutions to The Set of Linear Differential
Equations
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Linear Combinations of SolutionsRemember, For Linear
Equations, if
is a solution and
is a solution, then
is also a solution.
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Linear Combinations of Solutions
How can you tell?
Rememberand
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Linear Combinations of Solutions
How can you tell?
Rememberandso
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Linear Combinations of Solutions
How can you tell?
![Page 18: Homogeneous Linear Systems with Constant Coefficients Solutions of Systems of ODEs](https://reader034.vdocuments.us/reader034/viewer/2022052510/56649f3b5503460f94c5a2c2/html5/thumbnails/18.jpg)
Linear Combinations of Solutions
How can you tell?
Because
Are Scalars
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Linear Combinations of Solutions
How can you tell?
Because
Are Scalars
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Linear Combinations of Solutions
How can you tell?
So
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Fundamental Set of Solutions
Additionally, if the Eigenvectors are linearly independent
Then
Form a Fundamental Set of Solutions,
is the general solution.
det
and
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Why?Consider the Wronskian
det
Never 0
Only 0 if
are linearly dependent
det
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So To Solve
Determine All Eigenvalues and Eigenvectors of
General Solution takes the form
Plug in 0 and use initial conditions to find
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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.
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Questions?