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Ls5: Created page with "== Catalog Information == === Title === Mathematics for Engineering 2. === (Credit Hours:Lecture Hours:Lab Hours) === (4:4:0) === Offered === F, W === Prerequisite === [[Math..."

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Created page with "== Catalog Information == === Title === Mathematics for Engineering 2. === (Credit Hours:Lecture Hours:Lab Hours) === (4:4:0) === Offered === F, W === Prerequisite === [[Math..."

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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]] or [[Math 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]] or [[Math 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|Math 302 Mathematics for Engineering 2]]

← Older revision		Revision as of 21:56, 3 April 2013
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	== Catalog Information ==		== Catalog Information ==

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 === Prerequisite ===
-[[Math 302 Mathematics for Engineering 1|302]] or [[Math 314 Calculus of Several Variables:314]].
+[[Math 302 Mathematics for Engineering 1|302]] or [[Math 314 Calculus of Several Variables|314]].
 === Description ===
@@ Line 20: / Line 20: @@
 === Prerequisites ===
-Students are expected to have completed [[Math 302 Mathematics for Engineering 1|302]] or [[Math 314 Calculus of Several Variables:314]].
+Students are expected to have completed [[Math 302 Mathematics for Engineering 1|302]] or [[Math 314 Calculus of Several Variables|314]].
 === Minimal learning outcomes ===

← Older revision		Revision as of 21:48, 1 April 2013
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	=== Courses for which this course is prerequisite ===		=== Courses for which this course is prerequisite ===

−	[[Category:Courses\|~~302 Mathematics for Engineering 2~~]]	+	[[Category:Courses\|303]]

← Older revision		Revision as of 21:35, 1 April 2013
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	=== Courses for which this course is prerequisite ===		=== Courses for which this course is prerequisite ===

−	[[Category:Courses\|~~Math~~ 302 Mathematics for Engineering 2]]	+	[[Category:Courses\|302 Mathematics for Engineering 2]]