This exam will cover material from sections 1.1-1.3, 2.1-2.5, 3.1, 4.1, 4.3-4.4, 4.6-4.7, 5.1, and 7.1-7.5. Some material that covered in class and/or the homework is now excluded, namely:

• Direction Fields (all of 2.1)
• "Homogeneous" First Order Differential Equations (2.5)
• Euler's Method (all of 2.6)
• Existence and Uniqueness of Solutions (all of 2.7)
• Nonlinear 1st Order Modeling (all of 3.2)
• Reduction of Order (all of section 4.2)
• Electrical Circuits (parts of 3.1, 5.1)

At the time of the exam you will be provided with a table of Laplace Transforms, identical to the one given on the third test with the additional of Dirac delta function. As with the second test, you will also be provided a basic table for the method of undetermined coefficients, and the basic equations used for the method for variation of parameters

Generally, problems will fit into the following categories:

• Classification. Classify a differential equation in terms of linear/ nonlinear, order, and ODE/PDE
• Know the difference between the domain of a function and the interval of definition for a solution, and identify both
• First Order Differential Equations. Solve using separable, linear (integrating factor), or Bernoulli method. You should be able to identify the correct solution method if it is not stated.
• Applications to first order. You will be expected to find and solve the corresponding differential equation as part of each problem.
  • Motion
  • Mixing
  • Exponential growth (esp. population) and exponential decay (esp. half-life)
• Linear Dependence/Independence: definition for a set of solutions; how to determine if a set of vectors or functions is one or the other (e.g., Wronskian). Related to this is determining if a given set of functions form a fundamental set of solutions for an ODE.
• Homogeneous Second Order Differential Equations: constant coefficient; Euler-Cauchy; general solution; initial conditions
• Nonhomogeneous Second Order Differential Equations: constant coefficient only
• Harmonic Motion/Oscillations/Mass-Spring
• Laplace Transform/Inverse Laplace Transform
• Definition of the Laplace Transform: use to find the transform, including given only the graph of a function.
• Rewrite a piecewise continuous function in terms of unit step functions and find the Laplace Transform.
• Solve an initial value problem using Laplace Transforms.

There will be approximately 10-12 problems on the test. I will provide the paper and any additional scratch sheets you may need. Regular graphing and/or programmable calculators(e.g., TI-83/84/84+) may be used during the exam, but those that do symbolic computation (e.g., TI-89) are not permitted.