# Math 230 - Chapter 4 Study Guide

1. The roots of the auxiliary equation are given as well as the right hand side of the non-homogeneous equation. Write the complementary solution and the form of the particular solution. (3 parts)
2. Write an operator that will annihilate each expression. (3 parts)
3. Given the eqution using the D notation (example: D(D+1)y=0), write the general solution to the differential equation. (3 parts)
4. 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 differential equation. (2 parts).
5. Solve the homogeneous differential equation. This may be constant coefficients or Cauchy-Euler. (8 parts)
6. Solve the non-homogeneous differential equation.
7. Solve the system of linear differential equations.
8. You are given a system of differential equations. The x and y terms have been solved for using D notation. You will need to be able to write the complementary solution and find the particular solution. You will also need to rewrite the parameters on y in terms of the parameters on x.
9. Find the largest interval containing a point where the functions are linearly independent.
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. 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.
12. Use the substitution u = y' to solve the non-linear equation.

## 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.
• You may use the brief table of integrals from the front of your textbook.
• Many of the problems have been broken into smaller chunks rather than having you take a problem and work all the way through it. For example, in question 1, you're given the roots of the auxilliary equation and the form of the particular solution. So, instead of you being given y''+4y'+4y=e2x and you having to solve m2+4m+4=0 to get m=-2 with multiplicity 2, you're given L(y)=e2x and told m=-2, -2. Then, instead of having to find the particular solution, you're just asked to find the form of the particular solution.

## Points per problem

 # Pts 1 2 3 4 5 6 7 8 9 10 11 12 Total 9 9 9 6 24 5 5 6 4 6 12 5 100