Course Description: A study of the applications,
methods of solution, and basic theory of ordinary differential
equations(ODE). Classification of differential equations (order,
linearity, etc.). Solution of linear, exact, separable, and homogenous
first-order ODE. Numerical methods for solving ODE. Solution of
second-order and higher-order linear ODE with constant coefficients.
Series solutions of linear ODE with variable coefficients. Laplace
transform methods. Solution of systems of linear ODE. Qualitative analysis
of nonlinear ODE.
Course Objectives: To describe a variety of basic
methods for solving ordinary differential equations. To demonstrate the
usefulness of differential equations through applied problems. To increase
problem-solving skills and appreciation of the theory underlying the
techniques used. To provide an introduction to linear systems of linear
differential equations, meaning of eigenvalues and eigenvectors within the
context of systems of linear differential equations, and information
derived through linearization of nonlinear systems.
Course Outline by Topical Areas:
| Introduction to Differential Equations
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Definitions, Examples, Classification
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Integrating factor for Linear ODE
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Exact ODE, homogenous ODE
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Fundamental solution of Linear ODE
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Characteristic equation for linear ODE
with constant coefficients |
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General solution of homogeneous ODE
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| Second-order ODE, continued
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Particular solution of linear
non-homogeneous second order ODE |
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Method of undetermined coefficients
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Method of variation of parameters
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Higher order linear ODE with constant
coefficients |
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Non-linear and variable coefficient ODE
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Mechanical vibrations and other
applications |
| Systems of linear first-order ODE
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Coupled mass-spring systems, electrical
systems |
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Numerical methods for systems
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| Systems of non-linear ODE
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| Linear ODE with variable coefficients,
series solutions |
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Review of power series, analytic functions
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Ordinary and singular points
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Series solutions at an ordinary point
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| Series solutions at singular points
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Cauchy-Euler (equidimensional) equations
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Method of Frobenius (series solution at
regular singular point) |
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Finding the second solution, and some
special functions |
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Motivation: ODE with discontinuous forcing
terms (rhs) |
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Definition of Laplace transform
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Transforms of some simple functions
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| Laplace transforms, continued
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Laplace transforms of derivatives, and
other properties |
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Inverse Laplace transforms
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Basic procedure to solve ODE-IVP
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Transforms of discontinuous functions
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Impulses and the dirac delta function
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Solving linear systems using Laplace
transforms |
