SM212

Differential Equations

(4-0-4)

1. Text. Differential equations with BVPs 6th Zill and Cullen, Brooks-Cole.

2. Prerequisites.SM221 or SM223

3. Objectives

1. To provide the standard methods for solving differential equations as well as methods based on the use of matrices or Laplace transforms.
2. To demonstrate how differential equations can be useful in solving many types of problems - in particular, to show how to translate problems into the language of differential equations, to find or numerically approximate the solution of the resulting differential equation subject to given conditions, and to interpret the solutions obtained.
3. To study Fourier series and solve boundary values problems.
4. To develop numerical methods for solving differential equations.
5. To help prepare for the mathematics needed in the FE Exam (Fundamentals of Engineering).
4. Course Content - Class Hours (this is tentative - currently undergoing change)

1. Introduction - 2

2. First order differential equations - 4

3. Applications of first order differential equations - 2

4. Higher order linear differential equations - 6

5. Laplace transforms - 13

6. Applications of higher order differential equations - 7

7. Systems of differential equations - 6

8. Fourier Series and boundary-value problems - 8

9. Numerical methods of approximating solutions - 3

10. Tests - 4

11. Final Examination - 3

5. Acquired Abilities. Upon completion of this course, the student should be able to:

1. Classify differential equations in terms of the concepts: ordinary, partial, order, linear.

2. Verify that a relation is the general solution or a particular solution of a differential equation.

3. Find general (explicit and/or implicit) solutions to first order differential equations that are separable or linear.

4. Find solutions of these types subject to given conditions.

5. Use the direction field of a first order ordinary differential equation to plot solutions.

6. Superposition solutions to linear differential equations.

7. Set up, solve, and interpret first order differential equations arising in problems related to Newtonian mechanics, heat conduction, and fluid mixing.

8. Use operator notation.

9. Solve homogenous linear differential equations.

10. Solve nonhomogeneous differential equations by the method of undetermined coefficients.

11. Find Laplace transforms of given functions.

12. Find Laplace transforms using tables.

13. Find inverse Laplace transforms of functions.

14. Use Laplace transforms to solve linear differential equations.

15. Find inverse Laplace transforms using method of partial fractions and the method of convolution.

16. Solve vibrating spring problems with or without damping/external forces using standard methods and Laplace transform methods.

17. Solve vibrating spring problems without damping, with periodic external force, and determine if the behavior of pure resonance is present.

18. Solve electric circuit problems with or without charge/inductance/resistance/capacitance/electromotive force, using standard methods and Laplace transform methods.

Be able to recognize transient and steady state terms.
19. Use row-reduced echelon form (with the TI calculator rref command) to solve linear systems and be able to find inverses of matrices by row-reduction.

20. Use linear algebra (with suitable TI calculator commands) to eigenvalues and eigenvectors of matrices.

21. Solve systems of differential equations by matrix (i.e., eigenvalue) methods and by Laplace transforms.

22. Write a system of higher order equations as a system of first order equations and solve it numerically (using for example, Euler's method).

23. Solve electric networks using standard methods and Laplace transform methods.

24. Translate heat-flow problems into boundary-value problems.

25. Solve boundary-value problems using the method of separation of variables.

26. Obtain Fourier series and other orthogonal expansions of given functions.

27. Solve heat-flow problems using Fourier series.

Last updated 2011-09-18 by wdj.