First Order Linear Systems

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First Order Linear Systems

• MATH 224

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General Form of Linear DE Systems

• where
• X is a
• A is a
• F is a
• We'll start with homogeneous case,
• X'AX
• Solutions found using eigenvalues/vectors

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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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Distinct Complex Eigenvalues

• For complex eigenvalues, eigenvectors will also
  be complex
• Could make complex-looking solutions with
• We want real solutions

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Solutions from Complex Eigenvalues

• If
• ?abi, and
• VB1B2i for abi eigenvalue
• then two solutions to DE are

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Example 1 Complex Eigenvalues
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Example 1 Continued
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Example 2 3 Equations
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Example 2 - Continued
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Example 2 General Solution
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Example 2 Initial Conditions

• Adding initial conditions
• x(0) 3
• y(0) 0
• z(0) -1

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Continued
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Long-Term Behaviours

• In common with non-system DEs,

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Non-Homogeneous Case

• If we added external force to DE, we would get a
  non-homogeneous form,
• X'AX F
• Can be solved in manner similar to higher-order
  DEs, but with more exceptions, details
• More advanced methods work more generally not
  covered in this course, but in text

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Next steps

• For more on complex case, see pages 346-349
• Before next week, Read "Designing for
  Earthquakes", Project 8 page 365
• That will be our case study for next 2 classes,
  then final review class