Math 135 Notes, Winter 2015

You may download some notes that supplement the text here. They are pdf files.

1. Introduction to Higher-Order Linear Equations (5 January version)
   • Normal Forms and Solutions: Coefficients, Forcing.
   • Initial-Value Problems: Basic Existence and Uniqueness Theorem.
   • Intervals of Definition.
   • Overview of Higher-Order Linear Equations.

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

3. 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.

4. Nonhomogeneous Linear Equations: General Methods and Theory (5 January version)
   • Particular and General Solutions.
   • Solutions of Initial-Value Problems.

5. 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.

6. 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 Initial-Value Problems.
   • Piecewise-Defined Forcing.
   • Inverse Transform: Partial Fraction Decompositions.
   • Computing Green Functions.
   • Convolutions.
   • Impulse Forcing.

7. Theory for First-Order Equations (4 February version)
   • Well-Posed Initial-Value Problems: Notion of Well-Posedness, Classical Solutions of Initial-Value 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.

8. Theory for First-Order Systems (5 February version)
   • Normal Forms and Solutions.
   • Initial-Value Problems.
   • Recasting Higher-Order Problems as First-Order Systems.
   • Linear First-Order Systems.