Difference between revisions of "Math 303 Mathematics for Engineering 2"

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== Catalog Information ==
 
 
=== Title ===
 
Mathematics for Engineering 2.
 
 
=== (Credit Hours:Lecture Hours:Lab Hours) ===
 
(4:4:0)
 
 
=== Offered ===
 
F, W
 
 
=== Prerequisite ===
 
[[Math 302 Mathematics for Engineering 1|302]] or [[Math 314 Calculus of Several Variables|314]].
 
 
=== Description ===
 
ODEs, Laplace transforms, Fourier series, PDEs.
 
 
== Desired Learning Outcomes ==
 
This course is designed to give students from the College of Engineering and Technology the mathematics background necessary to succeed in their chosen field.
 
 
=== Prerequisites ===
 
Students are expected to have completed [[Math 302 Mathematics for Engineering 1|302]] or [[Math 314 Calculus of Several Variables|314]].
 
 
=== Minimal learning outcomes ===
 
Students should achieve mastery of the topics below.
 
<div style="-moz-column-count:2; column-count:2;">
 
# Some Basic Mathematical Models; Direction Fields
 
#* Model physical processes using differential equations.
 
#* Sketch the direction field (or slope field) of a differential equation using a computer graphing program.
 
#* Describe the behavior of the solutions of a differential equation by analyzing its slope field.  Identify any equilibrium  solutions.
 
# Solutions of Some Differential Equations; Classification of Differential Equations
 
#* Solve basic initial value problems; obtain explicit solutions if possible.
 
#* Characterize the solutions of a differential equation with respect to initial values.
 
#* Use the solution of an initial value problem to answer questions about a physical system.
 
#* Determine the order of an ordinary differential equation. Classify an ordinary differential equation as linear or nonlinear.
 
#* Verify solutions to ordinary differential equations.
 
#* Determine the order of a partial differential equation. Classify a partial differential equation as linear or nonlinear.
 
#* Verify solutions to partial differential equations.
 
# Linear First Order Equations with Variable Coefficients
 
#* Identify and solve first order linear equations.
 
#* Analyze the behavior of solutions.
 
#* Solve initial value problems for first order linear equations.
 
# Separable First Order Equations
 
#* Identify and solve separable equations; obtain explicit solutions if possible.
 
#* Solve initial value problems for separable equations, and analyze their solutions.
 
#* Apply the transformation $y=xv(x)$ to obtain a separable equation, if possible.
 
# Modeling with First Order Equations
 
#* Construct models of tank problems using differential equations.  Analyze the models to answer questions about the physical system modeled.
 
#* Construct growth and decay problems using differential equations.  Analyze the models to answer questions about the physical system modeled.
 
#* Construct models of problems involving force and motion using differential equations.  Analyze the models to answer questions about the physical system modeled.
 
#Differences Between Linear and Nonlinear Equations
 
#* Recall and apply the existence and uniqueness theorem for first order linear differential equations.
 
#* Recall and apply the existence and uniqueness theorem for first order differential equations (both linear and nonlinear).
 
#* Summarize the nice properties of linear equations. Contrast with nonlinear equations.
 
# Autonomous Equations and Population Dynamics
 
#* Determine and classify the equilibrium solutions of an autonomous equation as asymptotically stable, semistable or unstable by analyzing a graph of $\dfrac{dy}{dt}$ versus $y$. Sketch the phase line.
 
#* Analyze solutions of the logistic equation and other autonomous equations.
 
# Exact Equations and Integrating Factors
 
#* Identify whether or not a differential equation is exact.
 
#* Solve exact differential equations with or without initial conditions, and obtain explicit solutions if possible.
 
#* Use integrating factors to convert a differential equation to an exact equation and then solve.
 
#* Determine an integrating factor of the form $\mu(x)$ or $\mu(y)$ which will convert a non-exact differential equation to an exact equation, if possible.
 
# Introduction to Second Order Equations
 
#*  Determine the characteristic equation of a second order linear differential equation with constant coefficients.
 
#*  Solve second order linear differential equations with constant coefficients that have a characteristic equation with real  and distinct roots.
 
#*  Describe the behavior of solutions.
 
#*  Convert a second order differential equation to a first order differential equation in the following cases: i) y"=f(t,y'), ii) y"=f(y,y').
 
# Fundamental Solutions of Linear Homogeneous Equations; the Wronskian
 
#* Recall and apply the existence and uniqueness theorem for second order linear differential equations.
 
#* Recall and verify the principal of superposition for solutions of second order linear differential equations.
 
#* Evaluate the Wronskian of two functions.
 
#* Determine whether or not a pair of solutions of a second order linear differential equations constitute a fundamental set of solutions.
 
#* Recall and apply Abel's theorem.
 
# Complex Roots of the Characteristic Equation
 
#* Use Euler's formula to rewrite complex expressions in different forms.
 
#* Solve second order linear differential equations with constant coefficients that have a characteristic equation with complex roots.
 
#* Solve initial value problems and analyze the solutions.
 
# Repeated Roots; Reduction of Order
 
#* Solve second order linear differential equations with constant coefficients that have a characteristic equation with repeated roots.
 
#* Solve initial value problems and analyze the solutions.
 
#* Apply the method of reduction of order to find a second solution to a given differential equation.
 
# Nonhomogeneous Equations; Method of Undetermined Coefficients
 
#* For a nonhomogeneous second order linear differential equation, determine a suitable form of a particular solution that can be used in the method of undetermined coefficients.
 
#* Apply the method of undetermined coefficients to solve nonhomogeneous second order linear differential equations.
 
#* Solve initial value problems and analyze the solutions.
 
# Variation of Parameters; Reduction of Order
 
#* Apply the method of variation of parameters to solve nonhomogeneous second order linear differential equations with or without initial conditions.
 
#* Apply the method of reduction of order to solve nonhomogeneous second order linear differential equations with or without initial conditions.
 
# Mechanical Vibrations
 
#*  Model undamped mechanical vibrations with second order linear differential equations, and then solve.  Analyze the solution.  In particular, evaluate the frequency, period, amplitude, phase shift, and the position at a given time.
 
#* Model damped mechanical vibrations with second order linear differential equations, and then solve.  Analyze the solution.  In particular, evaluate the quasi frequency, quasi period, and the behavior at infinity.
 
#* Define critically damped and overdamped. Identify when these conditions exist in a system.
 
# Forced Vibrations
 
#* Model forced, undamped mechanical vibrations with second order linear differential equations, and then solve.  Analyze the solution.
 
#* Describe the phenomena of beats and resonance. Determine the frequency at which resonance occurs.
 
#* Model forced, damped mechanical vibrations with second order linear differential equations, and then solve.  Determine and analyze the solutions, including the steady state and transient parts.
 
# General Theory of nth Order Linear Equations
 
#* Recall and apply the existence and uniqueness theorem for higher order linear differential equations.
 
#* Recall the definition of linear independence for a finite set of functions.  Determine whether a set of functions is linearly independent or linearly dependent.
 
#* Use the Wronskian to determine if a set of solutions form a fundamental set of solutions.
 
#* Recall the relationship between Wronskian and linear independence for a set of functions, and for a set of solutions.
 
#* Apply the method of reduction of order to solve higher order linear differential equations.
 
# Homogeneous Equations with Constant Coefficients
 
#* Apply Euler's formula to write a complex number in exponential form.  Find the distinct complex roots of a number.
 
#* Determine the characteristic equation of  higher order linear differential equations.
 
#* Solve higher order linear differential equations.
 
#* Solve initial value problems.
 
# The Method of Undetermined Coefficients
 
#* For a nonhomogeneous higher order linear differential equation, determine a suitable form of a generalized particular solution that can be applied in the method of undetermined coefficients.
 
#* Use the method of undetermined coefficients to solve nonhomogeneous higher order linear differential equations.
 
#* Solve initial value problems.
 
# The Method of Variation of Parameters
 
#* Use the method of variation of parameters to solve nonhomogeneous higher order linear differential equations.
 
#* Solve initial value problems.
 
# Review of Power Series
 
#* Determine the radius of convergence of a power series.
 
#* Find the power series expansion of a function.
 
#* Manipulate expressions involving summation notation. Change the index of summation.
 
# Series Solutions near an Ordinary Point, Part I
 
#* Find the general solution of a differential equation using power series.
 
#* Solve initial value problems.  Analyze the solution.
 
# Series Solutions near an Ordinary Point, Part II
 
#* Given an initial value problem, use the differential equation to inductively determine the terms in the power series of the solution, expanded about the initial value.
 
#* Determine a lower bound for the radius of convergence of a series solution.
 
# Euler Equations
 
#* Find the general solution to an Euler equation in the cases of real distinct roots, equal roots, and complex roots.
 
#* Solve initial value problems for Euler equations and analyze their solutions.
 
# Definition of Laplace Transform
 
#* Sketch a piecewise defined function.  Determine if it is continuous, piecewise continuous or neither.
 
#* Evaluate Laplace transforms from the definition.
 
#* Determine whether an infinite integral converges or diverges.
 
# Solution of Initial Value Problems
 
#* Evaluate inverse Laplace transforms.
 
#* Use Laplace transforms to solve initial value problems.
 
#* Evaluate Laplace transforms using the derivative identity given in Problem 28 (p. 322) of the textbook.
 
# Step Functions
 
#* Sketch the graph of a function that is defined in terms of step functions.
 
#* Convert piecewise defined functions to functions defined in terms of step functions and vice versa.
 
#* Find the Laplace transform of a piecewise defined function.
 
#* Apply the shifting theorems (Theorems 6.3.1 and 6.3.2) to evaluate Laplace transforms and inverse Laplace transforms.
 
# Differential Equations with Discontinuous Forcing Functions
 
#*  Use Laplace transforms to solve differential equations with discontinuous forcing functions.
 
#* Analyze the solutions of differential equations with discontinuous forcing functions.
 
# Impulse Functions
 
#* Define an idealized unit impulse function.
 
#* Use Laplace transforms to solve differential equations that involve impulse functions.
 
#* Analyze the solutions of differential equations that involve impulse functions.
 
# The Convolution Integral
 
#* Evaluate the convolution of two functions from the definition.
 
#* Prove and disprove properties of the convolution operator.
 
#* Evaluate the Laplace transform of a convolution of functions.
 
#* Use the convolution theorem to evaluate inverse Laplace transforms.
 
#* Solve initial value problems using convolution.
 
# Introduction to Systems of First Order Equations
 
#* Transform a higher order linear differential equation into a system of first order linear equations.
 
#* Transform a system of first order linear equations to a single higher order linear equation.
 
#* Model physical systems that involve more than one unknown function with a system of differential equations.
 
#* Recall and apply methods of linear algebra.
 
# Basic Theory of Systems of First Order Linear Equations
 
#* Recall and verify the superposition principle for first order linear systems.
 
#* Relate the Wronskian to linear independence and a fundamental set of solutions.
 
# Homogeneous Linear Systems with Constant Coefficients
 
#* Sketch a direction field and a phase portrait for a homogeneous linear system with constant coefficients.
 
#* Find the general solution of a homogeneous linear system with constant coefficients in the case of real, distinct eigenvalues.
 
#* Determine if the origin is a saddle point or a node for a homogeneous linear system.  Classify a node as asymptotically stable or unstable.
 
#* Find general solutions, solve initial value problems, and analyze their solutions.
 
# Complex Eigenvalues
 
#* Sketch a direction field and a phase portrait for a homogeneous linear system with constant coefficients.
 
#* Find the general solution of a homogeneous linear system with constant coefficients in the case of complex eigenvalues.
 
#* Classify the origin as a saddle point, a node, a spiral point or a center.
 
#* Solve and analyze physical problems modeled by systems of differential equations.
 
# Fundamental Matrices
 
#* Determine a fundamental matrix for a system of equations.
 
#* Solve initial value problems using a fundamental matrix.
 
#* Prove identities using fundamental matrices.
 
# Repeated Eigenvalues
 
#* Sketch a direction field and a phase portrait for a homogeneous linear system with constant coefficients.
 
#* Find the general solution of a homogeneous linear system with constant coefficients in the case of repeated eigenvalues.
 
#* Identify improper nodes.  Classify them as asymptotically stable or unstable.
 
#* Solve initial value problems.
 
# Nonhomogeneous Linear Systems
 
#* Use diagonalization to solve nonhomogeneous linear systems.
 
#* Use the method of undetermined coefficients to solve nonhomogeneous linear systems.
 
#* Use the method of variation of parameters to solve nonhomogeneous linear systems.
 
#* Solve initial value problems.
 
# Two-Point Boundary Value Problems
 
#* Solve boundary value problems involving linear differential equations.
 
#* Find the eigenvalues and the corresponding eigenfunctions of a boundary value problem.
 
# Fourier Series
 
#* Identify functions that are periodic.  Determine their periods.
 
#* Find the Fourier series for a function defined on a closed interval.
 
#* Determine the $m$th partial sum of the Fourier series of a function.  Compare to the function.
 
# The Fourier Convergence Theorem
 
#* Find the Fourier series for a periodic function.
 
#* Recall and apply the convergence theorem for Fourier series.
 
# Even and Odd Functions
 
#* Determine whether a given function is even, odd or neither.
 
#* Sketch the even and odd extensions of a function defined on the interval [0,L].
 
#* Find the Fourier sine and cosine series for the function defined on [0,L].
 
#* Establish identities involving infinite sums from Fourier series.
 
# Separation of Variables; Heat Conduction in a Rod
 
#* Apply the method of separation of variables to solve partial differential equations, if possible.
 
#* Find the solutions of heat conduction problems in a rod using separation of variables.
 
# Other Heat Conduction Problems
 
#* Solve steady state heat conduction problems in a rod with various boundary conditions.
 
#* Analyze the solutions.
 
# The Wave Equation; Vibrations of an Elastic String
 
#* Solve the wave equation that models the vibration of a string with fixed ends.
 
#* Describe the motion of a vibrating string.
 
# Laplace's Equation
 
#* Solve Laplace's equation over a rectangular region for various boundary conditions.
 
#* Solve Laplace's equation over a circular region for various boundary conditions.
 
 
</div>
 
 
=== Textbooks ===
 
Possible textbooks for this course include (but are not limited to):
 
 
*
 
 
=== Additional topics ===
 
 
=== Courses for which this course is prerequisite ===
 
 
[[Category:Courses|303]]
 

Latest revision as of 15:15, 3 April 2013

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