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