Study Guide for Final/a>
This exam will cover material from sections 4.1-4.4, and 4.6-4.7.
Most questions will require solving a given differential equation, with or
without initial values, but not all (see, for example, sections 4.1,
4.2). Among other things, make sure that you understand what a fundamental
set of solutions is, what linear independent/dependent solutions are
(definition), and the relationship between the Wronskian and linear
independence. All equations included will be linear, and fit into one of the
following categories:
- Homogeneous with Constant Coefficients. Solved by assuming that
each solution has the general form y=erx and then solving the
auxiliary equation. Equations may be second, third or fourth order.
- Euler-Cauchy. Solve by assuming each solution has the general form
y=xm and then solving the auxiliary equation. Equations will only
be second order.
- In the case where one solution is known for a 2nd-order equation,
the reduction of order method is used to find the second solution. The
equation does not need to have constant coefficients. If a question of this
type is asked, you will be given at least one intermediate step.
- Nonhomogeneous with Constant Coefficients. Solved by
finding the homogeneous solution (by the methods above) and a particular
solution by the method of undetermined coefficients or variation of
parameters. Note that variation of parameters will only be used for
second-order equations.
In the case of homogeneous constant coefficient linear differential equations,
if you find a repeated root of the auxiliary equation, you may write down the
complete solution directly without proving it (similarly for complex roots).
In the case of a model for harmonic motion (e.g., a mass-spring system), you
would be given the governing ordinary differential equation. However, besides
solving the related differential equation or initial value problem, you will
likely be asked to comment on the solution (e.g., existence and type of
dampening, steady-state vs transient solutions, period, equilibrium position,
etc).
I will provide the paper and any additional scratch sheets you may need, along
with a short table for undetermined coefficients (p144, Table 4.1) and the
basic forms used with variation of parameters (essentially equations (5) on
p158). Calculator rules are the same as for the first exam (TI-89 type
calculators not allowed, but other scientific and graphing calculators may be
used).