Hi there —

The final grades are posted.

The average score on the online final for all 2214 students at Virginia Tech was a 72.9% (10.94 out of 15).

You all did much better: your average was an 80.3% (12.04 out of 15) — almost a whole letter grade higher! I appreciate the extra work you put in over the course of the semester; you made me look good.

Have a fantastic summer. It’s been an honor to work with you.

Heath

Quiz 8 will take place at the end of class on Thursday. Topics can include:

• Solve a system of first-order linear equations with complex eigenvalues.
• Understand the difference between ‘algebraic multiplicity’ and ‘geometric multiplicity’, and identify when a system is defective.
• Solve a defective system with a repeated real eigenvalue.
• Use the method of undetermined coefficients to solve a nonhomogeneous linear system.
• Use Euler’s method to numerically approximate the solution to a linear system.

Also, the extra credit project has been posted. The project has two parts, each worth two points for a maximum of four possible points. The project is optional, but if you choose to complete it, read the honor policy before beginning. One part is due by April 30, and the other part is due by May 3.

Feel free to omit 4.9 #6 from Homework 12. It’s not hard, but we didn’t get to it last week.

Quiz 7 will take place during the last 20 minutes of class on Thursday, and will cover questions from sections 4.1–4.5.

• Set up a two-tank mixing problem.
• Convert an nth-order linear differential equation into an n × n linear first-order system.
• Find the eigenvalues of a matrix.
• For a 3 × 3 matrix, find an eigenvector if the eigenvalues are already given.
• Completely solve a 2 × 2 linear, homogeneous initial value problem.

Quiz 6 will cover sections 3.11 and 3.12:

• Use synthetic division to factor a characteristic polynomial. (Example: Find the roots of )
• Find the general solution to a homogeneous differential equation of order > 2. (Example: Solve )
• Test a set of three functions to see if they form a fundamental set or not.
• Be able to write the general solution to a homogeneous differential equation with repeated complex roots. (Example: Solve using the fact that the characteristic polynomial factors as .)
• Calculate the nth roots of a complex number (such as the cube roots of ).

Homework 8 is posted.

Quiz 5 will take place on Thursday in class. The quiz will cover sections 3.6-3.8 and may include questions on these topics:

• Set up a spring problem involving unforced vibrations (example: an object weighing 9 pounds stretches a spring 6 inches at equilibrium. Write the differential equation modelling the motion of the spring.)
• Given an already-set up spring problem, identify whether the motion is overdamped, critically damped, or underdamped.
• Given a spring problem with initial conditions; find the solution equation in amplitude/phase-shift form.
• Use the equation to identify the parts of a solution to a nonhomogeneous differential equation.
• Use the Method of Undetermined Coefficients to identify the form of a particular solution to a nonhomogeneous differential equation.
• Solve a nonhomogeneous differential equation using the Method of Undetermined Coefficients.

Quiz 5 was scheduled for this Thursday (March 18), and no quiz was scheduled for the following Thursday (March 25). I’m moving Quiz 5 until March 25 to give folks a chance to work with the material and ask questions on it.

I’ll post more information about the quiz along with the next homework assignment.

Quiz 4 will take place during the last 15 minutes of class on Thursday. The format and length will be similar to other quizzes, but the topics will come from sections 3.1–3.4 and can include:

• Know the two criteria that must be satisfied for two functions for form a fundamental set.
• Use the Wronskian to test two functions for linear independence.
• Given a second order, linear, homogeneous differential equation whose characteristic function has real roots, identify whether the solution approaches zero as
• Given a second order, linear, homogeneous initial value problem whose characteristic function has real roots, find a particular solution.
• Know the form of the general solution when a characteristic equation has a repeated real root.
• You will not have to perform a Reduction of Order problem start to finish, but you should know the form of the substitution that allows the reduction of order to work.

Hi, folks:

I apologize for the very short notice, but I’m cancelling the 3:30 class and tonight’s review session due to a family emergency.

Unfortunately, I am not able to extend the deadline for this week’s Emporium quiz. That quiz covers sections 3.1 and 3.2, which we’ve already been over (it was the material on Homework 5). I’m cancelling Homework 6 this weekend to give people additional time to prepare for the quiz.

Again, I apologize for the short notice; thank you all for understanding.

Heath

Quiz 3 will take place during the last 15 minutes of class on Thursday. The format and length will be similar to Quiz 1, but the topics will come from sections 2.5, 2.6, 2.9 and 2.10, and can include:

• Solve a nonlinear equation using separation of variables.
• Solve a drag force problem when the drag force is proportional to velocity.
• Use the formula to find the terminal velocity of a falling point mass.
• Set up a drag force problem when the drag force is proportional to the square of the velocity (be careful about the signs!)
• Given a solution to an autonomous differential equation and an initial condition, find the solution that satisfies the initial condition.
• Use the substitution to solve a differential equation
• Use Euler’s Method to approximate the solution to a differential equation.