A Video Introduction to
Ordinary Differential Equations

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About This Course

These videos were recorded during  Professor Arthur Mattuck's class at MIT in the Spring of 2004. They are all approximately 50 min. in length.

 

Video 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 methodsBernouilli and homogeneous ODE's.
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Lecture 5 First-order autonomous ODE'squalitative methods, applications.
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Lecture 6 Complex numbers and complex exponentials.
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Lecture 7 First-order linear with constant coefficientsbehavior of solutions, use of complex methods.
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Lecture 8 Continuation; applications to temperature, mixing, RC-circuit, decay, and growth models.
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Lecture 9 Solving second-order linear ODE's with constant coefficientsthe three cases.
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Lecture 10 Continuation; complex characteristic roots;  undamped and damped oscillations.
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Lecture 11 Theory of general second-order linear homogeneous ODE'ssuperposition, uniqueness, Wronskians.
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Lecture 12 Continuation; general theory for inhomogeneous ODE's. Stability criteria for the constant-coefficient ODE's.
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Lecture 13 Finding particular solutions to inhomogeneous ODE'soperator and solution formulas involving exponentials.
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Lecture 14 Interpretation of the exceptional caser esonance.
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Lecture 15 Introduction to Fourier series; basic formulas for period 2(pi).
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Lecture 16 Continuationmore general periods; even and odd functions; periodic extension.
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Lecture 17 Finding particular solutions via Fourier series; resonant terms;hearing musical sounds.
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Lecture 19 Introduction to the Laplace transform; basic formulas.
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Lecture 20 Derivative formulas; using the Laplace transform to solve linear ODE's.
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Lecture 21 Convolution formulaproof, connection with Laplace transform, application to physical problems.
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Lecture 22  Using Laplace transform to solve ODE's with discontinuous inputs.
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Lecture 23 Use with impulse inputs; Dirac delta function, weight and transfer functions.
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Lecture 24 Introduction to first-order systems of ODE's; solution by elimination, geometric interpretation of a system.
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Lecture 25 Homogeneous linear systems with constant coefficients  solution via matrix eigenvalues (real and distinct case).
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Lecture 26 Continuationrepeated real eigenvalues, complex eigenvalues.
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Lecture 27 Sketching solutions of 2x2 homogeneous linear system with constant coefficients.
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Lecture 28 Matrix methods for inhomogeneous systemstheory, fundamental matrix, variation of parameters.
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Lecture 29 Matrix exponentials; application to solving systems.
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Lecture 30 Decoupling linear systems with constant coefficients.
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Lecture 31 Non-linear autonomous systemsfinding the critical points and sketching trajectories;  the non-linear pendulum.
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Lecture 32 Limit cycles existence and non-existence criteria.
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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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