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−  == Catalog Information ==
 
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−  === Title ===
 
−  Mathematics for Engineering 2.
 
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−  === (Credit Hours:Lecture Hours:Lab Hours) ===
 
−  (4:4:0)
 
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−  === Offered ===
 
−  F, W
 
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−  === Prerequisite ===
 
−  [[Math 302 Mathematics for Engineering 1302]] or [[Math 314 Calculus of Several Variables314]].
 
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−  === Description ===
 
−  ODEs, Laplace transforms, Fourier series, PDEs.
 
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−  == 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.
 
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−  === Prerequisites ===
 
−  Students are expected to have completed [[Math 302 Mathematics for Engineering 1302]] or [[Math 314 Calculus of Several Variables314]].
 
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−  === Minimal learning outcomes ===
 
−  Students should achieve mastery of the topics below.
 
−  <div style="mozcolumncount:2; columncount: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 nonexact 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.
 
−  # TwoPoint 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.
 
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−  </div>
 
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−  === Textbooks ===
 
−  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:Courses303]]
 