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Math 391, Section R Exam 1 Se
Fourier Series and the Dot Product
By Eli Amzallag
Inspired by Readings in Linear Algebra and
Better Dierential Equations Books Than B& D
Given a periodic function f (x) with exactly one period occurr
The Necessary Formulas : An Application of Cramers Rule
Examples
Application to First-Order
Generalization to Higher-Order Equations
Outline
1
The Necessary Formulas : An Application of Cramers Rule
2
The Necessary Formulas : An Application of Cramers Rule
Examples
Application to First-Order
Generalization to Higher-Order Equations
Math 39100: Methods of Differential Equations
Variation of Paramete
Second-Order LHODEs
Cramers Rule and the Wronskian
Existence-Uniqueness Theorem
Repeated Root Case and Reduction of Order
Eulers Formula and Complex Roots Case
An Example of Cramers Rule
3x + 5y = 1
2
Second-Order LHODEs
Cramers Rule and the Wronskian
Existence-Uniqueness Theorem
Repeated Root Case and Reduction of Order
Eulers Formula and Complex Roots Case
Observation if Wronskian is Nonzero at a
Second-Order LHODEs
Cramers Rule and the Wronskian
Existence-Uniqueness Theorem
Repeated Root Case and Reduction of Order
Eulers Formula and Complex Roots Case
The Guess
For reasons we will see later,
Second-Order LHODEs
Cramers Rule and the Wronskian
Existence-Uniqueness Theorem
Repeated Root Case and Reduction of Order
Eulers Formula and Complex Roots Case
Proof
Proof.
Suppose the vectors are lin
Second-Order LHODEs
Cramers Rule and the Wronskian
Existence-Uniqueness Theorem
Repeated Root Case and Reduction of Order
Eulers Formula and Complex Roots Case
Example 1 : IVP in Case 1
y y = 0, y (0)
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December 2,
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November 4,
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Exam 1
September 3
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