The following is a list of the major topics which will be covered on this exam. It is not presented in the same order as the textbook.

Topics:

Here are more detailed descriptions of some of these ideas and concepts.

(1) Generating and intepreting a mathematical model for a specific physical situation is one of the more difficult topics we considered.Most of the types of problems we considered led to simple separable or linear equations. You should know how to solve an acceleration-velocity problem with or without air resistance and understand the different possibilities for modelling air resistance, i.e., linearly or quadratically proportional to the velocity.

(2) The two numerical methods we studied were Euler's method and the Runge-Kutta method. You should be able to use these methods to derive approximations to solutions using a calculator. I will only ask you to iterate the method one or two times. Due to its complexity, I will provide you with the formulas for the Runge-Kutta method.

(3) Recognize general homogeneous and nonhomogeneous linear equations. The general solution to an nth-order homogeneous linear equation is a linear combination of n linearly independent functions. Know what it means for a collection of functions to be linearly dependent or linearly independent and be able to use the Wronskian to test this.

Know how to solve constant coefficient equations using the characteristic equation. Remember that there are various cases, according to the multiplicity of the roots of the characteristic equation and whether those roots are real or complex.

The general solution y(x) to a nonhomogeneous linear equation is the sum of the complementary solution y_c(x), which is the general solution to the associated homogeneous equation, and one particular solution y_p(x) to the given equation. We discussed two methods for finding the particular solution y_p(x): the method of undetermined coefficients and the method of variation of parameters. The former is easier to apply but only works in certain special cases; the latter works all the time but reduces the problem to the computation of a (possibly intractable) integral. Regarding the method of undetermined coefficients: know how to choose the right guess for y_p(x). It should be a linear combination of all of the terms which appear in the forcing term of the original differential equation or any of its derivatives. The situation is somewhat more complicated if any of these terms satisfy the associated homogeneous equation; then it is necessary to multiply all of the terms in your guess by a suitable power of x.

Here are a few sample problems for you to try:

1. Find the general solution to y'' + 3y' - 10y = 0

2. Solve the initial value problem 4y'' - y = 0, y(0) = 4, y'(0) = -1

3. Find the general solution to y'' - 6y' + 9y = 0

4. Solve the initial value problem y''' +25y' = 0, y(0) = 3, y'(0) = -5, y''(0) = 50

5. Find a particular solution to x'' + x' + x = cos t

6. Find the general solution to x'' + 4x' + 3x = 16e^t

7. Find the general solution to x'' + 4x' - 5x = e^t + 5t + 6

8. Solve the initial value problem x''' + 3x'' + 3x' + x = 5e^(-2t), x(0) = 1, x'(0) = 1, x''(0) = 1

(4) You should be able to apply these theoretical results to specific applied problems: vibrating springs, oscillating pendulums, buoyancy problems, etc. Understand the meaning of the term simple harmonic motion, and be prepared for problems involving damping terms.