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 equation 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. Find the largest interval containing a point where the functions are linearly independent.
6. Solve the homogeneous differential equation. This may be constant coefficients or Cauchy-Euler. (7 parts)
7. A differential equation is given in annihilator format 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.
8. Solve the system of linear differential equations. Combine parameters from x(t) and y(t) so that only c₁ and c₂ are used.
9. The Wronskian and associated determinants for L(y)=g(x) are given. Use them to find the complementary solution, g(x), the annihilator for the corresponding homogeneous differential equation, the annihilator for the non-homogeneous differential equation and the functions u₁', u₂', and u₃'. In addition, write the original equation in Lagrange's notation (y, y', y'', ...). You do not need to actually find for the particular solution.

Notes

• This exam uses the linear operator form to write a generic differential equation L(y) = g(x) without having actually give the differential equation. It follows that L(y_c) would mean the left-hand side when you plug in the complementary solution and L(y_p) would be the left-hand side when you plug in the particular solution.
• 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 m²+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.

Points per problem