First Order Linear System Theory

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First Order Linear System Theory

• MATH 224

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Last Class

• We saw how to convert higher-order DEs to a set
  (system) of first-order DEs
• We saw how to get MATLAB to predict numerically
  the solution of the DE
• Today we look at analytic solutions to linear
  systems of DEs
• generalization of earlier DE solutions

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Looking for analytic solutions
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General Form of Linear DE Systems

• where
• X is a
• A is a
• F is a

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Example 1
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Example 2
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Homogeneous Case

• Homogenous means F(t) 0
• DE becomes simply X' AX
• How many solutions, what form?
• If there are n variables, DE will have n linearly
  independent solutions
• any such set makes up a fundamental set of
  solutions

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General Solution Homog's

• If we can find X1, X2, , Xn vectors of functions
  which are solutions,
• General solution to X' A X will be

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How to find n solutions?

• Assume form for X(t)
• Sub into DE

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Eigenvalue/Eigenvector Eqn
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Example
Look up 'eig' function in MATLAB A 2 2 1 3
l eig(A) v, l eig(A) -0.8944 -0.7071
0.4472 -0.7071
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Cases

• Eigenvalues are real, distinct
• Eigenvalues are real, repeated
• Eigenvalues are distinct complex pair
• Eigenvalues are repeated complex pair

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Case 1 From Example

• For each real distinct eigenvalue ?, and
  corresponding eigenvector V, we get one solution
  vector, X1

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General Solution to Example
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Case 3 Distinct Complex Pairs

• Will look at next class
• Read 8.2, can skip "Repeated Eigenvalues" topic