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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]] or [[Math 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]] or [[Math 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="mozcolumncount:2; columncount: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 nonexact 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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−  # TwoPoint 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:CoursesMath 302 Mathematics for Engineering 2]]
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