Exam 4 Practice Problems

You will have to do four similar problems in class for Exam IV.  You can check some of your answers (and more) using the Maple Worksheet links for each problem.

1.  y" - 4y = 0       Solve using power series.  Find the recurrence formula and determine the series solution in terms of the sum of two series, one of which represents a_ocosh(2x) and the other of which represents (1/2)a₁sinh(2x).

Here is a Maple worksheet investigating series solutions to  y" - 4y = 0,  y(0) = 0,  y'(0) = 2.

2.  x²y" - 3xy' + 3y = 2x⁴e^x,  x > 0     (This is a Cauchy-Euler equation)

3.  (1 + x²)y" - 4xy' + 6y = 0       Solve using power series.  You should find that a_n = 0 for all n > 3.  In this problem you have no initial conditions so you are finding the general solution.  Your answer will be in terms of a_o and a₁.  (a_o and a₁ play the role of the arbitrary constants in this problem just as they do in problem 1.)  Maple worksheet

4.  xy" - y = 0,    y(2) = 0,   y'(2) = 3        Find the recurrence formula and give the minimum interval of convergence for the power series solution.  We will look at an Excel spreadsheet in class indicating y-solution values for various values of v (v = x - 2).  We will discuss them as they relate to the interval of convergence of the power series solution.  Include in your work on this problem the approximate solutions for y we arrive at in class corresponding to v-values of 0.5, 1, 1.5, 1.75, 1.9, 1.95, 1.99.  This Maple worksheet should give you some additional information.  Here is the Excel Worksheet I went over in class.  Each change column corresponds to the column directly to its left.

5.  2y" + xy' - 4y = x,    y(0) = 1,    y'(0) = -1/3     Maple Worksheet

6.  y''' - 2x²y" + 3xy' - 2y = 0,     y(0) = 0,    y'(0) = 0,   y"(0) = 2     Maple Worksheet

7.  y'' + xy' + y = sin(x),     y(0) = 0,     y'(0) = 1       Find the first six nonzero terms in the power series solution to this initial value problem.  You must show the work, including the computation of the necessary derivatives, in justifying your answer.  You can check your final result by looking at this Maple worksheet.

8.  Non-Linear Pendulum Problem--Click Here    You may complete this problem using Maple in the classroom (S219) with my help.

Recurrence Formulas for 1, 3-6, and Solution for 2

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        Lane Vosbury, Mathematics, Seminole State College   email:  vosburyl@seminolestate.edu

        This page was last updated on 08/21/14          Copyright 2002          webstats