Differential Equations

Course Description

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 (ODEs) 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.

Lectures

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   Lecture 1 - The Geometrical View of y'=f(x,y): Direction Fields, Integral Curves

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   Lecture 2 - Euler's Numerical Method for y'=f(x,y) and its Generalizations

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   Lecture 3 - Solving First-order Linear ODE's; Steady-state and Transient Solutions

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   Lecture 4 - First-order Substitution Methods: Bernouilli and Homogeneous ODE's

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   Lecture 5 - First-order Autonomous ODE's: Qualitative Methods, Applications

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   Lecture 6 - Complex Numbers and Complex Exponentials

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   Lecture 7 - First-Order Linear with Constant Coefficients

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   Lecture 8 - Applications to Temperature, Mixing, RC-circuit, Decay, and Growth Models

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   Lecture 9 - Solving Second-Order Linear ODE's with Constant Coefficients

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    Lecture 10 - Complex Characteristic Roots; Undamped and Damped Oscillations

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    Lecture 11 - Second-Order Linear Homogeneous ODE's: Superposition, Uniqueness, Wronskians

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    Lecture 12 - Inhomogeneous ODE's; Stability Criteria for Constant-Coefficient ODE's

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    Lecture 13 - Inhomogeneous ODE's: Operator and Solution Formulas Involving Ixponentials

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    Lecture 14 - Interpretation of the Exceptional Case: Resonance

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    Lecture 15 - Introduction to Fourier Series; Basic Formulas for Period 2(pi)

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    Lecture 16 - More General Periods; Even and Odd Functions; Periodic Extension

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    Lecture 17 - Finding Particular Solutions via Fourier Series; Resonant Terms

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    Lecture 18 - Derivative Formulas; Using the Laplace Transform to Solve Linear ODE's

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    Lecture 19 - Convolution Formula: Proof, Connection with Laplace Transform, Application

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    Lecture 20 - Using Laplace Transform to Solve ODE's with Discontinuous Inputs

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    Lecture 21 - Impulse Inputs; Dirac Delta Function, Weight and Transfer Functions

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    Lecture 22 - First-Order Systems of ODE's; Solution by Elimination, Geometric Interpretation

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    Lecture 23 - Homogeneous Linear Systems with Constant Coefficients: Solution via Matrix Eigenvalues

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    Lecture 24 - Continuation: Repeated Real Eigenvalues, Complex Eigenvalues

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    Lecture 25 - Sketching Solutions of 2x2 Homogeneous Linear System with Constant Coefficients

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    Lecture 26 - Matrix Methods for Inhomogeneous Systems

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    Lecture 27 - Matrix Exponentials; Application to Solving Systems

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    Lecture 28 - Decoupling Linear Systems with Constant Coefficients

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    Lecture 29 - Non-linear Autonomous Systems: Finding the Critical Points and Sketching Trajectories

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    Lecture 30 - Limit Cycles: Existence and Non-existence Criteria

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    Lecture 31 - Non-Linear Systems and First-Order ODE's