21-260: Differential Equations

Course Overview

Section 1.1: Prologue: What are Differential Equations?

Section 1.2: Four Introductory Models.

Radioactive Decay/Carbon Dating

Newton's Law of Cooling

Spread of Disease

Section 1.3: Fundamental Concepts and Terminology.

Order, Linearity, Homogeneity

Superposition

General and Particular Solutions

Initial Value Problems

Section 2.1: Methods of Solution (Linear Equations).

Integrating Factors

Initial Value Problems

Section 2.2.2: Mixing Problems.

Mathematical Modeling

Constant/variable volume

Section 3.1: Direction Fields and Numerical Approximation.

Direction Fields

Isoclines

Euler's Method

Section 3.2: Separable Equations.

Solving separable equations.

Section 3.5.2: Nonlinear First-order Equations in Applications.

Toricelli's Law

Section 4.2: Existence and Uniqueness.

unique local solution, continuation, maximal solution

Existence & Uniqueness Theorem

Consequences of Uniqueness

Section 4.3: Qualitative and Asymptotic Behavior.

Phase Line

Stability of Equilibrium Points

Asymptotic behavior

Section 4.4: The Logistic Population Model.

Model building

Carrying capacity, growth rate

Harvesting

Section 5.1: Introduction: Modeling Vibrations.

Mass-Spring systems

Damping

External Forces

Section 5.2: State Variables and Numerical Approximations.

State Space/Phase Plane

Existence and Uniqueness

Section 5.3: Operators and Linearity.

Differential Operators

Linearity

Superposition Principle

Section 5.4: Solutions and Linear Independence.

Linear Dependence and Independence

Structure of General Solutions

the Wronskian and linear indepencence

Section 6.1: Homogeneous Equations with Constant Coefficients.

Characteristic Equation

Exponential Solutions

Differential Operators

Section 6.2: Exponential Shift.

Repeated Roots of the characteristic equation

Exponential Forcing Functions

Section 6.3: Complex Roots.

Euler-DeMoivre Formula

Complex Roots of the characteristic equation.

Section 6.4: Real Solutions from Complex Solutions.

Superposition Principle

Sinusoidal Forcing Functions

Section 6.5: Unforced Vibrations.

Amplitude, angular frequency, phase shift.

Overdamping, Critical Damping, Underdamping

Section 6.6: Periodic Force and Responce.

Resonance!

Gain

Damped Forced Equations

Section 7.1: Definition and Pasic Properties (of the Laplace Transform).

Definition

First Differentiation Theorem

Inverse Transform

Partial Fractions

Section 7.2: More Transforms and Further Properties.

First Shift Theroem

Second Differentiation Theorem

Section 7.3: Heavyside Functions and Piecewise-Defined Inputs.

Second Shift Theorem

Section 7.4: Periodic Inputs.

Periodic Extensions

Transforms fo Periodic Functions

Section 7.5: Impulses and the Dirac Distribution.

Impulse

Laplace Transform of the Dirac Distribution

Section 7.6: Convolution.

Convolution Theorem

Finding Inverse Transforms - Green's Functions

Section 11.1: The Basic Diffusion Problem.

Partial Differential Equations

Initial Conditions and Boundary Conditions

The Heat Equation

Zero temperature endpoints

Insulated endpoints

Constant temperature endpoints

Section 11.2: Solutions by Separation of Variables.

u(t,x)=T(t)X(x)

second order equations for T(t) and X(x)

Boundary conditions

non-zero solutions

Linear Combinations

Section 11.3: Fourier Series.

Fourier Coefficients

Convergence

Section 11.4: Fourier Sine and Cosine Series.

Even and odd functions

Even and odd extensions of a function

Fourier Sine and Cosine coefficitnts

Section 12.1: The Wave Equation.

Separation of Variables

Non-zero Initial position

Non-zero initial velocity

Harmonics

Energy