# Math 230 - Chapter 4 Study Guide

1. The roots of the auxiliary equation are given. Write the complementary solution and the form of the particular solution. (3 parts)
2. Write an operator that will annihilate each expression. (4 parts)
3. Given the eqution using the D notation (example: D(D+1)y=0), write the general solution to the differential equation. (5 parts)
4. The general solution to a second order homogeneous linear differential equation is given. Identify whether the given conditions form an initial value problem or boundary value problem. Find the constants that satisfy the given conditions. (3 parts)
5. Find the largest interval containing a point where the functions are linearly independent.
6. Use the method of undetermined coefficients to find the particular solution. The form of the particular solution is given along with the simplification from Maxima when the particular solutions is plugged into the equation.
7. Use the substitution to solve the equation.
8. Solve the homogeneous differential equation. This may be constant coefficients or Cauchy-Euler. (5 parts)
9. A non-homogeneous D.E. is given in linear format (example: L(y)=sin x) along with its Wronskian. Write an annihilator for the complementary function, write the complementary function, write an annihilator for the non-homogeneous equation, write the original equation in y, y', y'', ... format, and find the derivative functions for u1', u2', and u3' that would be used in the variation of parameters method. You do not need to actually solve for the particular solution.
10. The three solutions to the complementary function as well as the values of u1', u2', and u3' are given. Write the complementary solution and use variation of parameters to find the particular solution.
11. You are given a system of differential equations. Rewrite it in standard form using the D operator. The x and y terms have been eliminated for you, but you need to be able to interpret what remains.

## Notes

• You will need to bring scratch paper for extra work.
• You will not get to use Maxima during the exam. Most of the heavy calculations have been done for you. You will need a calculator.
• This will be a one day exam. If you don't know it, then you don't get extra time to finish it. Go through and work the easiest ones that are worth the most points first. If you get bogged down, move on to something else.

## Points per problem

 # Pts 1 2 3 4 5 6 7 8 9 10 11 Total 9 8 10 6 4 4 6 25 10 6 12 100