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Math 391, Section R Exam 1 September 3f}. BUM 1. Match the differeutial equations to
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 occurring on the interval
[L, L], we would like to write it a
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
Examples
3
Application to First-Order
4
Generalization
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 Parameters
Eli Amzallag
City College
1 / 22
The Necessary Formu
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
2x 6y = 10
The unique solution to this system is x = 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 Single Point
Suppose we have n solutions to an nth ord
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, we guess that a solution to
ay + by + cy = 0
has the f
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 linearly dependent. This means that
c1 v1 + c2 v2 = 0, whe
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) = 2, y (0) = 1
The characteristic equation is r 2 1 =
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Math 391, Section R
Exam 3
December 2, 2014
1. Consider the dierential equation (2 + x2 )y xy
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Math 391, Section R Exam 3 December 2, 2014 1. Consider the differential equation (
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Math 391, Section R Exam 2 November 4, 2014 1. Consider the differential equatio
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Math 391, Section R
Exam 2
November 4, 2014
1. Consider the dierential equation t2 y 4ty + 6y
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Math 391. Section 1 A Exam 3 July 22, 2013 1. Consider the differential eq iation {
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Math 391, Section 13m Exam 2 Jul}: 3, EDIE 1. Use your knewledge ef lineer indep
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Math 391, Section R
Exam 1
September 30, 2014
1. Match the dierential equations to the direct
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Math 391, Section 1XA
Exam 2
July 3, 2013
1. Use your knowledge of linear independence and