Math 135 Notes, Winter 2015
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They are pdf files.

Introduction to HigherOrder Linear Equations
(5 January version)
 Normal Forms and Solutions:
Coefficients, Forcing.
 InitialValue Problems:
Basic Existence and Uniqueness Theorem.
 Intervals of Definition.
 Overview of HigherOrder Linear Equations.

Homogeneous Linear Equations: General Methods and Theory
(5 January version)
 Linear Differential Operators:
Coefficients, Forcing.
 Method of Superposition:
Application to InitialValue Problems,
Genersal Initial Conditions.
 Wronskians: Abel Wronskian Theorem.
 Fundamental Sets of Solutions and General Solutions.
 Natrual Fundamental Sets of Solutions.
 Linear Independence of Solutions.

Homogeneous Linear Equations with Constant Coefficients
(5 January version)
 Characteristic Polynomials and the Key Identity.
 Real Roots of Characteristic Polynomials:
Simple Real Roots, Real Roots with any Multiplicity.
 Complex Extension of the Key Identity.
 Complex Roots of Characteristic Polynomials:
Simple Complex Roots, Complex Roots with any Multiplicity.

Nonhomogeneous Linear Equations: General Methods and Theory
(5 January version)
 Particular and General Solutions.
 Solutions of InitialValue Problems.

Nonhomogeneous Linear Equations with Constant Coefficients
(5 January version)
 Key Identity Evaluations: Setting Up Key Identity Evaluations,
Zero Degree Examples, Positive Degree Examples, Why It Works.
 Undetermined Coefficients: Form for Particular Solutions,
Determining the Undetermined Coefficients, Examples, Why It Works.
 Forcings of Composite Characterisitc Form.
 Green Functions for Constant Coefficient Equations:
Examples, Why It Works.

Laplace Transform Method
(4 February version)
 Definition of the Transform: Examples.
 Properties of the Transform: Linearity,
Exponentials and Translations, Heaviside Function.
 Existence and Differentiablity of the Transform:
Piecewise Continuity, Exponential Order,
Existence and Differentiablity.
 Transform of Derivatives.
 Application to InitialValue Problems.
 PiecewiseDefined Forcing.
 Inverse Transform: Partial Fraction Decompositions.
 Computing Green Functions.
 Convolutions.
 Impulse Forcing.

Theory for FirstOrder Equations
(4 February version)
 WellPosed InitialValue Problems: Notion of WellPosedness,
Classical Solutions of InitialValue Problems.
 Linear Equations: Existence and Uniqueness Theorem,
Intervals of Definition.
 Separable Equations: Recipe for Solutions,
Nonuniqueness of Solutions, Existence and Uniqueness Theorem.
 General Equations: Picard Existence and Uniqueness Theorem,
Integral Formulation, Gronwall Lemma and Uniqueness,
Picard Iteration.

Theory for FirstOrder Systems
(5 February version)
 Normal Forms and Solutions.
 InitialValue Problems.
 Recasting HigherOrder Problems as FirstOrder Systems.
 Linear FirstOrder Systems.