This exam will cover material from sections 1.1-1.3, 2.1-2.3, 2.5-2.6, and 3.1-3.2. Specifically excluded is material relating to:

• Verify that a unique solution exists to an IVP (1.2)
• Autonomous differential equations (2.1)

The majority of questions will require solving a differential equation, with or without initial values. If the problem states that a particular solution method is used, you must use that method; some problems may not indicate which method to use. You should be also able to identify and describe the solution method needed to solve a particular equation without necessarily solving the equation. There will be at least one application (possibly more) involving modeling.

Other problem types that may appear include the following:

• 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
• Draw a solution curve on a direction field, given the inital condition
• Approximate the solution to a differential equation at a given value using Euler's Method (given the general Euler's formula for y_n+1); understand the relationship with errors in the method, and be able to calculate the absolute and relative errors

1. Separable. Used when dy/dx = h(x)g(y).
2. Linear. Used when the equation can be written in the form dy/dx + p(x)y = g(x); solve with an integrating factor.
3. Bernoulli. Used when the equation can be written in the form dy/dx + p(x)y = g(x)yⁿ; solve by substituting u=y^1-n in order to get a linear equation in u.
4. Other Substitutions. Given a substitution, be able to covert a differential equation into a linear or separable equation.

I will provide the paper and any additional scratch sheets you may need. Graphing or programmable calculators may be used during the exam, BUT you must show all your work by hand. Solutions based on results found by calculators will not be accepted.