Course Features
1 The Geometrical View of y'=f(x,y): Direction Fields, Integral Curves.
2 Euler's Numerical Method for y'=f(x,y) and its Generalizations.
3 Solving First-order Linear ODE's; Steady-state and Transient Solutions.
4 First-order Substitution Methods: Bernouilli and Homogeneous ODE's.
5 First-order Autonomous ODE's: Qualitative Methods, Applications.
6 Complex Numbers and Complex Exponentials.
7 First-order Linear with Constant Coefficients: Behavior of Solutions, Use of Complex Methods.
8 Continuation; Applications to Temperature, Mixing, RC-circuit, Decay, and Growth Models.
9 Solving Second-order Linear ODE's with Constant Coefficients: The Three Cases.
10 Continuation: Complex Characteristic Roots; Undamped and Damped Oscillations.
11 Theory of General Second-order Linear Homogeneous ODE's: Superposition, Uniqueness, Wronskians.
12 Continuation: General Theory for Inhomogeneous ODE's. Stability Criteria for the Constant-coefficient ODE's.
13 Finding Particular Sto Inhomogeneous ODE's: Operator and Solution Formulas Involving Ixponentials.
14 Interpretation of the Exceptional Case: Resonance.
15 Introduction to Fourier Series; Basic Formulas for Period 2(pi).
16 Continuation: More General Periods; Even and Odd Functions; Periodic Extension.
17 Finding Particular Solutions via Fourier Series; Resonant Terms; Hearing Musical Sounds.
19 Introduction to the Laplace Transform; Basic Formulas.
20 Derivative Formulas; Using the Laplace Transform to Solve Linear ODE's.
21 Convolution Formula: Proof, Connection with Laplace Transform, Application to Physical Problems.
22 Using Laplace Transform to Solve ODE's with Discontinuous Inputs.
23 Use with Impulse Inputs; Dirac Delta Function, Weight and Transfer Functions.
24 Introduction to First-order Systems of ODE's; Solution by Elimination, Geometric Interpretation of a System.
25 Homogeneous Linear Systems with Constant Coefficients: Solution via Matrix Eigenvalues (Real and Distinct Case).
26 Continuation: Repeated Real Eigenvalues, Complex Eigenvalues.
27 Sketching Solutions of 2x2 Homogeneous Linear System with Constant Coefficients.
28 Matrix Methods for Inhomogeneous Systems: Theory, Fundamental Matrix, Variation of Parameters.
29 Matrix Exponentials; Application to Solving Systems.
30 Decoupling Linear Systems with Constant Coefficients.
31 Non-linear Autonomous Systems: Finding the Critical Points and Sketching Trajectories; the Non-linear Pendulum.
32 Limit Cycles: Existence and Non-existence Criteria.
33 Relation Between Non-linear Systems and First-order ODE's; Structural Stability of a System, Borderline Sketching Cases; Illustrations Using Volterra's Equation and Principle.