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Lectures on Differential Equations, MIT-Spring 2006, Prof. Arthur Mattuck


Lecture 1-The Geometrical View of y^{\prime}=f(x,y): Direction Fields, Integral Curves.
Lecture 2-Euler’s Numerical Method for y^{\prime}=f(x,y) and its Generalizations.
Lecture 3-Solving First-order Linear ODE’s; Steady-state and Transient Solutions.
Lecture 4-First-order Substitution Methods: Bernouilli and Homogeneous ODE’s.
Lecture 5-First-order Autonomous ODE’s: Qualitative Methods, Applications.
Lecture 6-Complex Numbers and Complex Exponentials.
Lecture 7-First-order Linear with Constant Coefficients: Behavior of Solutions, Use of Complex Metho
Lecture 8-Continuation; Applications to Temperature, Mixing, RC-circuit, Decay, and Growth Models.
Lecture 9-Solving Second-order Linear ODE’s with Constant Coefficients: The Three Cases
Lecture 10-Continuation: Complex Characteristic Roots; Undamped and Damped Oscillations.
Lecture 11-Theory of General Second-order Linear Homogeneous ODE’s: Superposition, Uniqueness, Wronskians.
Lecture 12-Continuation: General Theory for Inhomogeneous ODE’s. Stability Criteria for the Constant-coefficient ODE’s.
Lecture 13-Finding Particular Sto Inhomogeneous ODE’s: Operator and Solution Formulas Involving Exponentials.
Lecture 14-Interpretation of the Exceptional Case: Resonance.
Lecture 15-Introduction to Fourier Series; Basic Formulas for Period 2\pi.
Lecture 16-Continuation: More General Periods; Even and Odd Functions; Periodic Extension.
Lecture 17-Finding Particular Solutions via Fourier Series; Resonant Terms; Hearing Musical Sounds.
Lecture 18-missing !!!
Lecture 19-Introduction to the Laplace Transform; Basic Formulas.
Lecture 20-Derivative Formulas; Using the Laplace Transform to Solve Linear ODE’s.
Lecture 21-Convolution Formula: Proof, Connection with Laplace Transform, Application to Physical Problems.
Lecture 22-Using Laplace Transform to Solve ODE’s with Discontinuous Inputs.
Lecture 23-Use with Impulse Inputs; Dirac Delta Function, Weight and Transfer Functions
Lecture 24-Introduction to First-order Systems of ODE’s; Solution by Elimination, Geometric Interpretation of a System.
Lecture 25-Homogeneous Linear Systems with Constant Coefficients: Solution via Matrix Eigenvalues (Real and Distinct Case).
Lecture 26-Continuation: Repeated Real Eigenvalues, Complex Eigenvalues.
Lecture 27-Sketching Solutions of 2\times 2 Homogeneous Linear System with Constant Coefficients
Lecture 28-Matrix Methods for Inhomogeneous Systems: Theory, Fundamental Matrix, Variation of Parameters.
Lecture 29-Matrix Exponentials; Application to Solving Systems.
Lecture 30-Decoupling Linear Systems with Constant Coefficients.
Lecture 31-Non-linear Autonomous Systems: Finding the Critical Points and Sketching Trajectories; the Non-linear Pendulum.
Lecture 32-Limit Cycles: Existence and Non-existence Criteria.
Lecture 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.


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