Engineering Differential Equations 92.236-201

Spring 2015

Course Materials

Exam Review Sheets, Including Practice Exams

1. Review Sheet for Exam 1


2. Review Sheet for Exam 2


3. Review Sheet for Exam 3


4. Review Sheet for Final Exam

Solution Sheets for Practice Exams

1. Solutions to Practice Exam 1


2. Solutions to Practice Exam 2


3. Solutions to Practice Exam 3


4. Solutions to Practice Final Exam

Take-Home Parts of Exams

1. Take-home Part of Exam 1


2. Take-home Part of Exam 2


3. Take-home Part of Exam 3

Solution Sheets for Exams

1. Solutions to Exam 1 Problems


2. Solutions to Exam 2 Problems


3. Solutions to Exam 3 Problems

Project

Homework for Sections 2.4, 2.6, 4.3

1. Section 2.4 homework handout


2. Section 2.6 homework handout


3. Section 4.3 homework handout

MATLAB Handouts

1. Handout on Graphing (January 22).


2. Handout on Symbolic Utilities (January 29).


3. Handout on Numerical Methods (February 12). You will also need Dr. White's scripts euler.m and rk4.m and the homework assignments for section 2.4 and section 2.6.


4. Handout on Slope Fields (February 19). You will also need Dr. John Polking's script dfield8.m. In case you don't have your textbook with you, the homework assignment on slope fields includes problems 1, 2, and 7from section 1.3.


5. Handout on additional MATLAB graphics commands (February 26).


6. Handout on Phase Planes (March 12). You will also need Dr. John Polking's script pplane8.m


7. Handout on Numerical Methods for Systems (March 26) to be used for the section 4.3 homework. You will also need Dr. White's script rk4.m.


8. Handout on Symbolic Laplace Transform Utilities (April 16).


9. Handout on Matrix Algebra Commands

Class Handouts

1. Solution method for first order linear equations


2. Solving first order equations


3. Solving second order constant coefficient equations


4. The Method of Undetermined Coefficients


5. Example of the Method of Variation of Parameters


6. Examples of forced motion


7. Example of forced motion with discontinuous forcing term


8. Example of forced motion with an impulse force

Challenge Problems

1. A friend of mine in college claimed that hot water freezes faster than cold water. His reasoning was that hot water loses heat at a faster rate than cold water, so if you put a container of hot water and a container of cold water in a freezer at the same time, the hot water will cool down faster and will freeze first. What does our heating/cooling model have to say about my friend's theory?


2. The Snowplow Problem: One night it started snowing at a steady rate. A snowplow started out at midnight. At 1 a.m., it had gone 2 miles. At 2 a.m., it had gone another mile. What time did it start snowing?


3. An interesting application of radioactive dating.


4. The standard form of a second-order linear d.e. is y″ + p(x) y′ + q(x) y = f(x). Recalling the solution procedure for a first-order linear d.e., you might think that there is an integrating factor ρ(x) with the property that, if you multiply both sides of the d.e. by ρ(x), the left side becomes the second derivative of ρ(x)y.
   • Show that this is possible only if q(x)=½ p′(x) + ¼ [p(x)]²
   • Find a formula for ρ(x) if q(x)=½ p′(x) + ¼ [p(x)]²


5. For people who have studied linear algebra:
   • Show that the set of solutions to the linear homogeneous d.e. y'' + p(x) y' + q(x) y = 0 is a subspace of the set of all twice differentiable functions.
   
   
   • Let y₁ denote the solution of the IVP y'' + p(x) y' + q(x) y = 0, y(0) = 1, y'(0) = 0 and let y₂ denote the solution of the IVP y'' + p(x) y' + q(x) y = 0, y(0) = 0, y'(0) = 1. (Theorem 2 in section 3.1 of Edwards & Penney's text guarantees that each of these IVP's has a unique solution on any interval on which the functions p and q are continuous.) Show that the set {y₁ , y₂} is a basis for the subspace in part a. Hint: Show that {y₁ , y₂} is linearly independent, and show that {y₁ , y₂} spans the subspace by expressing the solution of the IVP y'' + p(x) y' + q(x) y = 0, y(0) = a, y'(0) = b as a linear combination of y₁ and y₂.


6. The total energy E of a mass-spring system is given by E = ½mv² + ½kx². Show that E is constant for an undamped, unforced system, and show that E is a decreasing function of time for a damped, unforced system.