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

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== Catalog Information ==
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=== Title ===
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Mathematics for Engineering 2.
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=== (Credit Hours:Lecture Hours:Lab Hours) ===
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(4:4:0)
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=== Offered ===
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F, W
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=== Prerequisite ===
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[[Math 302 Mathematics for Engineering 1|302]] or [[Math 314 Calculus of Several Variables|314]].
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=== Description ===
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ODEs, Laplace transforms, Fourier series, PDEs.
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== Desired Learning Outcomes ==
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This course is designed to give students from the College of Engineering and Technology the mathematics background necessary to succeed in their chosen field.
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=== Prerequisites ===
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Students are expected to have completed [[Math 302 Mathematics for Engineering 1|302]] or [[Math 314 Calculus of Several Variables|314]].
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=== Minimal learning outcomes ===
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Students should achieve mastery of the topics below.
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<div style="-moz-column-count:2; column-count:2;">
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# Some Basic Mathematical Models; Direction Fields
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#* Model physical processes using differential equations.
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#* Sketch the direction field (or slope field) of a differential equation using a computer graphing program.
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#* Describe the behavior of the solutions of a differential equation by analyzing its slope field.  Identify any equilibrium  solutions.
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# Solutions of Some Differential Equations; Classification of Differential Equations
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#* Solve basic initial value problems; obtain explicit solutions if possible.
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#* Characterize the solutions of a differential equation with respect to initial values.
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#* Use the solution of an initial value problem to answer questions about a physical system.
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#* Determine the order of an ordinary differential equation. Classify an ordinary differential equation as linear or nonlinear.
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#* Verify solutions to ordinary differential equations.
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#* Determine the order of a partial differential equation. Classify a partial differential equation as linear or nonlinear.
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#* Verify solutions to partial differential equations.
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# Linear First Order Equations with Variable Coefficients
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#* Identify and solve first order linear equations.
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#* Analyze the behavior of solutions.
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#* Solve initial value problems for first order linear equations.
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# Separable First Order Equations
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#* Identify and solve separable equations; obtain explicit solutions if possible.
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#* Solve initial value problems for separable equations, and analyze their solutions.
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#* Apply the transformation $y=xv(x)$ to obtain a separable equation, if possible.
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# Modeling with First Order Equations
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#* Construct models of tank problems using differential equations.  Analyze the models to answer questions about the physical system modeled.
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#* Construct growth and decay problems using differential equations.  Analyze the models to answer questions about the physical system modeled.
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#* Construct models of problems involving force and motion using differential equations.  Analyze the models to answer questions about the physical system modeled.
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#Differences Between Linear and Nonlinear Equations
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#* Recall and apply the existence and uniqueness theorem for first order linear differential equations.
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#* Recall and apply the existence and uniqueness theorem for first order differential equations (both linear and nonlinear).
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#* Summarize the nice properties of linear equations. Contrast with nonlinear equations.
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# Autonomous Equations and Population Dynamics
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#* 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.
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#* Analyze solutions of the logistic equation and other autonomous equations.
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# Exact Equations and Integrating Factors
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#* Identify whether or not a differential equation is exact.
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#* Solve exact differential equations with or without initial conditions, and obtain explicit solutions if possible.
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#* Use integrating factors to convert a differential equation to an exact equation and then solve.
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#* 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.
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# Introduction to Second Order Equations
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#*  Determine the characteristic equation of a second order linear differential equation with constant coefficients.
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#*  Solve second order linear differential equations with constant coefficients that have a characteristic equation with real  and distinct roots.
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#*  Describe the behavior of solutions.
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#*  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').
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# Fundamental Solutions of Linear Homogeneous Equations; the Wronskian
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#* Recall and apply the existence and uniqueness theorem for second order linear differential equations.
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#* Recall and verify the principal of superposition for solutions of second order linear differential equations.
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#* Evaluate the Wronskian of two functions.
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#* Determine whether or not a pair of solutions of a second order linear differential equations constitute a fundamental set of solutions.
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#* Recall and apply Abel's theorem.
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# Complex Roots of the Characteristic Equation
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#* Use Euler's formula to rewrite complex expressions in different forms.
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#* Solve second order linear differential equations with constant coefficients that have a characteristic equation with complex roots.
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#* Solve initial value problems and analyze the solutions.
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# Repeated Roots; Reduction of Order
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#* Solve second order linear differential equations with constant coefficients that have a characteristic equation with repeated roots.
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#* Solve initial value problems and analyze the solutions.
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#* Apply the method of reduction of order to find a second solution to a given differential equation.
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# Nonhomogeneous Equations; Method of Undetermined Coefficients
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#* 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.
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#* Apply the method of undetermined coefficients to solve nonhomogeneous second order linear differential equations.
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#* Solve initial value problems and analyze the solutions.
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# Variation of Parameters; Reduction of Order
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#* Apply the method of variation of parameters to solve nonhomogeneous second order linear differential equations with or without initial conditions.
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#* Apply the method of reduction of order to solve nonhomogeneous second order linear differential equations with or without initial conditions.
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# Mechanical Vibrations
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#*  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.
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#* 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.
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#* Define critically damped and overdamped. Identify when these conditions exist in a system.
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# Forced Vibrations
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#* Model forced, undamped mechanical vibrations with second order linear differential equations, and then solve.  Analyze the solution.
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#* Describe the phenomena of beats and resonance. Determine the frequency at which resonance occurs.
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#* 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.
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# General Theory of nth Order Linear Equations
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#* Recall and apply the existence and uniqueness theorem for higher order linear differential equations.
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#* Recall the definition of linear independence for a finite set of functions.  Determine whether a set of functions is linearly independent or linearly dependent.
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#* Use the Wronskian to determine if a set of solutions form a fundamental set of solutions.
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#* Recall the relationship between Wronskian and linear independence for a set of functions, and for a set of solutions.
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#* Apply the method of reduction of order to solve higher order linear differential equations.
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# Homogeneous Equations with Constant Coefficients
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#* Apply Euler's formula to write a complex number in exponential form.  Find the distinct complex roots of a number.
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#* Determine the characteristic equation of  higher order linear differential equations.
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#* Solve higher order linear differential equations.
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#* Solve initial value problems.
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# The Method of Undetermined Coefficients
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#* 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.
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#* Use the method of undetermined coefficients to solve nonhomogeneous higher order linear differential equations.
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#* Solve initial value problems.
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# The Method of Variation of Parameters
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#* Use the method of variation of parameters to solve nonhomogeneous higher order linear differential equations.
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#* Solve initial value problems.
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# Review of Power Series
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#* Determine the radius of convergence of a power series.
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#* Find the power series expansion of a function.
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#* Manipulate expressions involving summation notation. Change the index of summation.
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# Series Solutions near an Ordinary Point, Part I
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#* Find the general solution of a differential equation using power series.
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#* Solve initial value problems.  Analyze the solution.
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# Series Solutions near an Ordinary Point, Part II
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#* 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.
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#* Determine a lower bound for the radius of convergence of a series solution.
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# Euler Equations
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#* Find the general solution to an Euler equation in the cases of real distinct roots, equal roots, and complex roots.
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#* Solve initial value problems for Euler equations and analyze their solutions.
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# Definition of Laplace Transform
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#* Sketch a piecewise defined function.  Determine if it is continuous, piecewise continuous or neither.
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#* Evaluate Laplace transforms from the definition.
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#* Determine whether an infinite integral converges or diverges.
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# Solution of Initial Value Problems
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#* Evaluate inverse Laplace transforms.
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#* Use Laplace transforms to solve initial value problems.
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#* Evaluate Laplace transforms using the derivative identity given in Problem 28 (p. 322) of the textbook.
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# Step Functions
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#* Sketch the graph of a function that is defined in terms of step functions.
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#* Convert piecewise defined functions to functions defined in terms of step functions and vice versa.
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#* Find the Laplace transform of a piecewise defined function.
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#* Apply the shifting theorems (Theorems 6.3.1 and 6.3.2) to evaluate Laplace transforms and inverse Laplace transforms.
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# Differential Equations with Discontinuous Forcing Functions
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#*  Use Laplace transforms to solve differential equations with discontinuous forcing functions.
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#* Analyze the solutions of differential equations with discontinuous forcing functions.
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# Impulse Functions
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#* Define an idealized unit impulse function.
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#* Use Laplace transforms to solve differential equations that involve impulse functions.
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#* Analyze the solutions of differential equations that involve impulse functions.
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# The Convolution Integral
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#* Evaluate the convolution of two functions from the definition.
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#* Prove and disprove properties of the convolution operator.
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#* Evaluate the Laplace transform of a convolution of functions.
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#* Use the convolution theorem to evaluate inverse Laplace transforms.
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#* Solve initial value problems using convolution.
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# Introduction to Systems of First Order Equations
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#* Transform a higher order linear differential equation into a system of first order linear equations.
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#* Transform a system of first order linear equations to a single higher order linear equation.
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#* Model physical systems that involve more than one unknown function with a system of differential equations.
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#* Recall and apply methods of linear algebra.
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# Basic Theory of Systems of First Order Linear Equations
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#* Recall and verify the superposition principle for first order linear systems.
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#* Relate the Wronskian to linear independence and a fundamental set of solutions.
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# Homogeneous Linear Systems with Constant Coefficients
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#* Sketch a direction field and a phase portrait for a homogeneous linear system with constant coefficients.
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#* Find the general solution of a homogeneous linear system with constant coefficients in the case of real, distinct eigenvalues.
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#* Determine if the origin is a saddle point or a node for a homogeneous linear system.  Classify a node as asymptotically stable or unstable.
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#* Find general solutions, solve initial value problems, and analyze their solutions.
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# Complex Eigenvalues
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#* Sketch a direction field and a phase portrait for a homogeneous linear system with constant coefficients.
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#* Find the general solution of a homogeneous linear system with constant coefficients in the case of complex eigenvalues.
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#* Classify the origin as a saddle point, a node, a spiral point or a center.
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#* Solve and analyze physical problems modeled by systems of differential equations.
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# Fundamental Matrices
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#* Determine a fundamental matrix for a system of equations.
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#* Solve initial value problems using a fundamental matrix.
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#* Prove identities using fundamental matrices.
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# Repeated Eigenvalues
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#* Sketch a direction field and a phase portrait for a homogeneous linear system with constant coefficients.
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#* Find the general solution of a homogeneous linear system with constant coefficients in the case of repeated eigenvalues.
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#* Identify improper nodes.  Classify them as asymptotically stable or unstable.
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#* Solve initial value problems.
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# Nonhomogeneous Linear Systems
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#* Use diagonalization to solve nonhomogeneous linear systems.
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#* Use the method of undetermined coefficients to solve nonhomogeneous linear systems.
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#* Use the method of variation of parameters to solve nonhomogeneous linear systems.
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#* Solve initial value problems.
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# Two-Point Boundary Value Problems
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#* Solve boundary value problems involving linear differential equations.
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#* Find the eigenvalues and the corresponding eigenfunctions of a boundary value problem.
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# Fourier Series
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#* Identify functions that are periodic.  Determine their periods.
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#* Find the Fourier series for a function defined on a closed interval.
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#* Determine the $m$th partial sum of the Fourier series of a function.  Compare to the function.
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# The Fourier Convergence Theorem
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#* Find the Fourier series for a periodic function.
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#* Recall and apply the convergence theorem for Fourier series.
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# Even and Odd Functions
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#* Determine whether a given function is even, odd or neither.
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#* Sketch the even and odd extensions of a function defined on the interval [0,L].
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#* Find the Fourier sine and cosine series for the function defined on [0,L].
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#* Establish identities involving infinite sums from Fourier series.
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# Separation of Variables; Heat Conduction in a Rod
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#* Apply the method of separation of variables to solve partial differential equations, if possible.
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#* Find the solutions of heat conduction problems in a rod using separation of variables.
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# Other Heat Conduction Problems
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#* Solve steady state heat conduction problems in a rod with various boundary conditions.
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#* Analyze the solutions.
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# The Wave Equation; Vibrations of an Elastic String
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#* Solve the wave equation that models the vibration of a string with fixed ends.
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#* Describe the motion of a vibrating string.
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# Laplace's Equation
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#* Solve Laplace's equation over a rectangular region for various boundary conditions.
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#* Solve Laplace's equation over a circular region for various boundary conditions.
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</div>
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=== Textbooks ===
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Possible textbooks for this course include (but are not limited to):
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*
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=== Additional topics ===
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=== Courses for which this course is prerequisite ===
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[[Category:Courses|303]]
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Latest revision as of 15:15, 3 April 2013

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