| 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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