|
|
| Line 1: |
Line 1: |
| | {{db-g7}} | | {{db-g7}} |
| − | == 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]]
| |
