# Math 303: Math for Engineering 2

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## Contents

## Catalog Information

### Title

Mathematics for Engineering 2.

### (Credit Hours:Lecture Hours:Lab Hours)

(4:4:0)

### Offered

F, W

### Prerequisite

### Description

ODEs, Laplace transforms, Fourier series, PDEs.

## Desired Learning Outcomes

This course is designed to give students from the College of Engineering the mathematics background necessary to succeed in their chosen field.

### Prerequisites

Students are expected to have completed Math 302 or Math 314.

### Minimal learning outcomes

Students should achieve mastery of the topics below.

- 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.

### Textbooks

Possible textbooks for this course include (but are not limited to):