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 firstorder linear ODE's;
steadystate and transient solutions.
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Lecture 4 
Firstorder substitution
methodsBernouilli and homogeneous ODE's.
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Lecture 5 
Firstorder autonomous
ODE'squalitative methods, applications.
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Lecture 6 
Complex numbers and complex
exponentials.
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Lecture 7 
Firstorder linear with constant
coefficientsbehavior of solutions, use of complex methods.
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Lecture 8 
Continuation; applications to
temperature, mixing, RCcircuit, decay, and growth models.
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Lecture 9 
Solving secondorder 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 secondorder
linear homogeneous ODE'ssuperposition, uniqueness, Wronskians.
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Lecture 12 
Continuation; general theory for
inhomogeneous ODE's. Stability criteria for the constantcoefficient 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 firstorder 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 
Nonlinear autonomous
systemsfinding the critical points and sketching trajectories; the nonlinear
pendulum.
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Lecture 32 
Limit cycles existence and
nonexistence criteria.
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Lecture 33 
Relation between nonlinear systems
and firstorder ODE's; structural stability of a system, borderline sketching cases;
illustrations using Volterra's equation and principle.
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