Differential Equations are the language in which the laws of nature are expressed. Understanding properties of solutions of differential equations is fundamental to much of contemporary science and engineering. Ordinary differential equations (ODE's) deal with functions of one variable, which can often be thought of as time. Topics include: Solution of first-order ODE's by analytical, graphical and numerical methods; Linear ODE's, especially second order with constant coefficients; Undetermined coefficients and variation of parameters; Sinusoidal and exponential signals: oscillations, damping, resonance; Complex numbers and exponentials; Fourier series, periodic solutions; Delta functions, convolution, and Laplace transform methods; Matrix and first order linear systems: eigenvalues and eigenvectors; and Non-linear autonomous systems: critical point analysis and phase plane diagrams.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.
Video Lectures | MPEG4 Video 320x240 25.00fps | AAC 24000Hz stereo 768Kbps | 33 lectures, (40 :50) minutes/lecture | 3.1 GB
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