Exam #1 Review

Scheduling Information:

 Time: Wednesday, June 5, during your usual class time
Location: GHC 4215. (Our usual classroom.)
 

Topics Covered:

 Introduction to Differential Equations. (Section 1.1 and 1.2)
Separable Differential Equations. (Section 2.2)
Some Basic Mathematical Models. (Sections 1.3)
Linear First Order Differential Equations. (Section 2.3)
Existence and Uniqueness Theorems. (Section 1.2, Examples 4 and 5)
Autonomous Differential Equations. (Section 2.1.2, Section 3.1, and Section 3.2. )
Direction Fields. (Section 2.1.1)
Phase Line. (Section 2.1.2, Section 3.1, and Section 3.2.)
Population Modelling. (Sections 3.1 and 3.2)
Higher Order Linear Differential Equations. (Section 4.1)
Second Order, Linear, Homogeneous, Constant Coefficient, Differential Equations. (Section 4.3)
Non-homogeneous Equations and the Method of Undetermined Coefficients. (Section 4.4)
 

Review Questions:
  1. What is a differential equation?
  2. What is a solution to a differential equation?
  3. How can you determine if a particular function is a solution to a given differential equation?
  4. What is an ordinary differential equation (ODE)? A partial differential equation (PDE)?
  5. What is the order of a differential equation?
  6. What is a linear first order differential equation?
  7. What is a separable differential equation?
  8. How can you find the general solution to a separable differential equation? [1. constant solutions, 2. other solutions]
  9. What is the domain of a solution to an initial value problem?
  10. What are the three steps in building a mathematical model? [1. science (assumptions); 2. Notation & units; 3. mathematics (translate assumptions into a mathematical statement)]
  11. What are the assumptions that lead to Newton's Law of cooling?
  12. What is the differential equation that describes Newton's law of cooling?
  13. How to we find a mathematical model for mixing problems [ {stuff in}-{stuff out} ]
  14. What is a linear first order differential equation?
  15. What is an integrating factor for a linear first order differential equation? What does it allow us to do?
  16. How do we find the integrating factor?
  17. How do we find the solution? [I recommend multiplying the equation, integrating both sides, and solving for "y(x)" over memorizing a formula. Strongly!]
  18. What is the direction field for a differential equation?
  19. What are the isoclines of a first order differential equation
  20. How can a direction field give you information about solutions to a differential equation?
  21. What type of questions can a direction field help you answer? What type of question is it not so good at answering?
  22. What is meant by "existence and uniqueness" of solutions to an initial value problem?
  23. We have a theorem about existence and uniqueness for linear first order equations. What are the hypotheses of that theorem? What are the conclusions?
  24. We have another theorem about existence and uniqueness that applies to all first order equations. What are the hypotheses of that theorem? What are the conclusions?
  25. What do these theorems imply about the graphs of solutions in regions where the hypotheses are satisfied?
  26. What is an autonomous differential equation?
  27. What can be said about the isoclines of an autonomous equation?
  28. What is the phase line of an autonomous differential equation? What information does it show about the differential equation?
  29. What is the exponential growth model? On what assumptions is this model based? How does this model predict that populations will behave?
  30. What is the logistic growth model? On what assumptions is this model based? How does this model predict that populations will grow?
  31. Why is p(t)y''+q(t)y'+r(t)y=g(t) called a *linear* equation?
  32. Why is it linear even if p, q, and r may not be linear functions of t?
  33. What can be said about linear combinations of solutions of homogeneous linear equations?
  34. What does it mean for a set of functions to be linearly independent?
  35. What is the fundamental set of solutions if the characteristic polynomial has distinct real roots? The general solution?
  36. What is the fundamental set of solutions if the characteristic polynomial has repeated real roots? The general solution?
  37. What is the fundamental set of solutions if the characteristic polynomial has complex roots? The general solution?
  38. How can you find solutions to initial value problems using the general solution?
  39. What is a particular solution to a non-homogeneous linear equation? What is the complementary solution?
  40. How can you find the general solution to a non-homogeneous equation? Why is it only necessary to find one particular solution?
  41. What is the Method of Undetermined Coefficients? How is it used to find a particular solution?
 

Exercises:

 Section 1.1: #3, 5, 7, 11, 13, 17, 21, 23, 43, 51, 55
Section 1.3 #5, 9, 11, 13, 17.
Section 2.3 #3, 5, 7, 13, 17, 25, 27, 31, 35.
Section 1.2 #19, 21, 31, 47, 50.
Section 2.1 #1, 3, 15, 21, 25, 39,
Section 3.1 #3, 13, 17, 23, 25, 41, 42,
Section 3.2 #5(a), 21.
Section 4.1 #5, 7, 9, 15, 21, 23, 27.
Section 4.3 #1, 5, 7, 29, 35.
Section 4.4 #1, 5, 7, 13, 19, 23, 29, 43.
 

Old Exam Problems:

Here is a large collection of exam problems I've given to Differential Equations students in previous semesters. Not all the problems are appropriate for this exam. I would recommend the following problems: 1-3, 5-6, 10, 12-13, 15-28, 30-33, 35-40, 42-46, 48-49, 51, 53-54, 56-63, 65-69, 71, 73-75, 77-82. Some problems on this list are repeated. Sorry about that.

Here is rather smaller selection of old exam problems related to second order equations.

Before the questions come in - no, I don't have solutions available for these problems. You have a large number of problems from the text with answers, and it's important for you to get used to working on problems where the answers are not available. Your TAs and I will be happy to answer your questions about these problems.