- 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)
- Write an operator that will annihilate each expression. (3 parts)
- Given the equation using the D notation (example: D(D+1)y=0), write the general solution to the differential equation. (3 parts)
- 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).
- Solve the homogeneous differential equation. This may be constant coefficients or Cauchy-Euler. (8 parts)
- Find the largest interval containing a point where the functions are linearly independent.
- A differential equation is given in annihilator format as well as the values of u
_{1}', u_{2}', and u_{3}' are given. Write the complementary solution and use variation of parameters to find the particular solution. - Solve the system of linear differential equations.
- 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 Lagrange's notation (y, y', y'', ...), and find the derivative functions for u
_{1}', u_{2}', and u_{3}' that would be used in the variation of parameters method. You do not need to actually find for the particular solution. - Find the first four non-zero terms of the Maclaurin series solution to the given initial value problem.

- You may want to bring additional scratch paper.
- 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.
- 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=e
^{2x}and you having to solve m^{2}+4m+4=0 to get m=-2 with multiplicity 2, you're given L(y)=e^{2x}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.

# | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | Total |
---|---|---|---|---|---|---|---|---|---|---|---|

Pts | 12 | 9 | 9 | 6 | 32 | 3 | 6 | 5 | 12 | 6 | 100 |