# Math 303 Mathematics for Engineering 2

### Title

Mathematics for Engineering 2.

(4:4:0)

F, W

302 or 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 302 or 314.

### Minimal learning outcomes

Students should achieve mastery of the topics below.

1. 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.
2. 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.
3. 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.
4. 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.
5. 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.
6. 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.
7. 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.
8. 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.
9. 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').
10. 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.
11. 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.
12. 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.
13. 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.
14. 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.
15. 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.
16. 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.
17. 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.
18. 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.
19. 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.
20. 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.
21. 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.
22. 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.
23. 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.
24. 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.
25. 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.
26. 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.
27. 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.
28. 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.
29. 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.
30. 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.
31. 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.
32. 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.
33. 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.
34. 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.
35. Fundamental Matrices
• Determine a fundamental matrix for a system of equations.
• Solve initial value problems using a fundamental matrix.
• Prove identities using fundamental matrices.
36. 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.
37. 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.
38. 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.
39. 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.
40. The Fourier Convergence Theorem
• Find the Fourier series for a periodic function.
• Recall and apply the convergence theorem for Fourier series.
41. 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.
42. 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.
43. Other Heat Conduction Problems
• Solve steady state heat conduction problems in a rod with various boundary conditions.
• Analyze the solutions.
44. 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.
45. 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):