Ch 1.1: Basic Mathematical Models; Direction Fields
Differential equations are equations containing derivatives. The following are examples of physical phenomena involving rates of change:
Motion of fluids Motion of mechanical systems Flow of current in e
Ch 1.2: Solutions of Some Differential Equations
Recall the free fall and owl/mice differential equations:
v = 9.8 0.2v, p = 0.5 p 450
These equations have the general form y' = ay - b We can use methods of calculus to solve differential equations of this
MATH 219
Fall 2015
Lecture 9
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Lecture notes by Ozgr Kiisel
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Content: Constant coecient systems. Complex eigenvalues.
Suggested Problems:
7.6: 1a, 6a, 8, 10, 25
1
Constant Coecient Linear Systems (cont.)
Example 1.1 Solve the system
1 1 2
x = 0 2 2 x
1
MATH 219
Fall 2015
Lecture 7
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Lecture notes by Ozgr Kiisel
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Content: Systems of linear algebraic equations; Linear independence, eigenvalues,
eigenvectors.
Suggested Problems:
7.3: 2, 5, 9, 10, 13, 15, 19, 21, 22, 24
1
Matrices of Functions
We can work
MATH 219
Fall 2015
Lecture 11
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Lecture notes by Ozgr Kiisel
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Content: Nonhomogenous linear systems (variation of parameters only).
Suggested Problems:
7.9: 1, 3, 5, 10, 12, 13
1
Variation of Parameters
Suppose that we have a nonhomogenous linear system
MATH 219
Fall 2015
Lecture 12
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Lecture notes by Ozgr Kiisel
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Content: General theory of nth order linear equations.
Suggested Problems:
4.1: 7, 9, 14, 16, 25
1
Linear ODEs
Suppose that y is a dependent variable which will be a function of t. An nth ord
MATH 219
Fall 2015
Lecture 13
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Lecture notes by Ozgr Kiisel
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Content: Homogeneous equations with constant coecients.
Suggested Problems:
4.2: 12, 13, 18, 21, 22, 30, 33, 35
1
Homogenous equations with constant coecients
Consider now an ODE of the form
MATH 219
Fall 2015
Lecture 10 (part 1)
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Lecture notes by Ozgr Kiisel
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Content: Fundamental Matrices. Repeated eigenvalues.
Suggested Problems:
7.7: 5, 7, 9
7.8: 1c, 4c, 5, 8a, 16, 19, 20abc
1
Matrix Exponentials, Fundamental Matrices
Let A be a constan
MATH 219
Fall 2015
Lecture 14
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Lecture notes by Ozgr Kiisel
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Content: The method of undetermined coecients.
Suggested Problems:
4.3: 1, 3, 4, 7, 8, 10, 11, 12
1
Solving linear, non-homogenous ODEs
Suppose now that we have
y (n) + a1 y (n1) + . . . + an