Math 285 Exam #3 Study Guide
Spring 2012
5.1. Review of Power Series
 Be able to find interval and radius of convergence
 Know/Recognize common Taylor series like the exponential, natural log, sine/cosine, and
hyperbolic sine and cosine.
 Be able to write a rational function as a power series
 Be able to re-index a power series to start at a given value of n.
 Be familiar with (at least) the ratio test, the alternating series test and the p-series test.
5.2 Solutions near an Ordinary Point
 Be able to solve a second order equation with constant coefficients by means of a power series.
 Be able to test for whether the solutions you obtain are fundamental solutions
 Be able to set up second order nonlinear equations by means of a power series.
10.1 Two-Point Boundary-Value Problems
 Be able to solve second order boundary value problems with constant coefficients
 Be able to find the eigenvalues and eigenfunctions of linear second order boundary value
problems
10.2 Fourier Series
 Know the formulas for Fourier Series and Fourier coefficients
 Find the Fourier series for a given periodic function and be able to state its period
 Review Integration by Parts!
10.3 Fourier Convergence Theorem
 Be able to check the conditions under which a function can be represented by a Fourier Series
10.4 Even and Odd Functions
 Recognize/Test even and odd function properties
 Recognize how even-/oddness affects the coefficients of a Fourier series.
 Be able to extend a function from an interval (0,L) to (-L,L) according to even or odd properties
1. Determine the interval and radius of convergence.
( 1)n 24 n x n

n 0 (2n  1)!

( x  1)n
b. 
n!
n 1

a.

n
x
c.  2  
n 0  3 

(10n  3)( x  2)2 n 1
d. 
n 2n
n 1
2. Represent the function as a power series (or Taylor Series) centered around the given point.
Graph your function using the first 5 terms and compare to the original.
a. F(x)=cosh(x), x0=0
b. G(x)=sinh(x), x0=1
1
c. 𝑓(𝑥) = 1+𝑥 , 𝑥0 = 1
d. g(x) = sin(2x) , 𝑥0 = 0
𝑛
3. Given the power series 𝑦 = ∑∞
𝑛=0 𝑎𝑛 (𝑥 − 1) , write the first 4 derivatives as power series that
start at n=0.
4. Solve the equation by using power series centered around the given point. Determine if the
solution set is a fundamental one. Where possible, state what function the series represents.
a. 𝑦 ′′ − 𝑦 = 0, 𝑥0 = 0
b. 𝑦 ′′ + 4𝑦 = 0, 𝑥0 = 0
c. 𝑦 ′′ + 4𝑦 ′ + 4𝑦 = 0, 𝑥0 = 0
d. 𝑦 ′′ − 𝑦 = 0, 𝑥0 = 3
e. 𝑦 ′′ + 𝑥𝑦 ′ + 2𝑦 = 0, 𝑥0 = 0
f. 𝑥𝑦 ′′ + 𝑦 ′ + 𝑥𝑦 = 0, 𝑥0 = 1
5. Solve the boundary value problem or show that it has no solution. If there is a solution,
characterize it as trivial or nontrivial, finite/unique, infinite set of solutions. Is the problem
homogeneous or non-homogeneous?
a. 𝑦 ′′ + 𝑦 = 0, 𝑦(0) = 0, 𝑦(𝐿) = 0
b. 𝑦 ′′ + 2𝑦 = 0, 𝑦(0) = 1, 𝑦(𝐿) = 0
c. 𝑦 ′′ + 2𝑦 = 𝑥, 𝑦(0) = 0, 𝑦(𝜋) = 0
d. 𝑦 ′′ + 4𝑦 = cos(𝑥) , 𝑦(0) = 0, 𝑦′(𝜋) = 0
6. Find the eigenvalues and eigenfunctions for the boundary value problem.
a. 𝑦 ′′ + 𝜆𝑦 = 0, 𝑦(0) = 0, 𝑦 ′ (𝜋) = 0
b. 𝑦 ′′ + 𝜆𝑦 = 0, 𝑦 ′ (0) = 0, 𝑦 ′ (𝐿) = 0
c. 𝑦 ′′ + 2𝜆𝑦 ′ + 𝜆2 𝑦 = 0, 𝑦(0) = 0, 𝑦(𝐿) = 1
d. 𝑦 ′′ + 4𝑦 ′′ + 𝜆𝑦 = 0, 𝑦(0) = 0, 𝑦(𝐿) = 0
e. 𝑦 ′′ − 𝜆𝑦 = 0, 𝑦(0) = 0, 𝑦 ′ (𝐿) = 1
7. Find the Fourier series to represent the following functions. State the period of each. You may
need to extend the interval (you may do this by even or odd properties, or simply set the rest of
the interval equal to 0). Sketch the graph you obtain using the first 5 terms of the series.
1
−𝐿 ≤𝑥 <0
a. 𝑓(𝑥) = {
𝑓(𝑥 + 2𝐿) = 𝑓(𝑥)
0
0≤𝑥<𝐿
−1
−2 ≤ 𝑥 < 0
b. 𝑓(𝑥) = {
𝑓(𝑥 + 4) = 𝑓(𝑥)
1
0≤𝑥<2
𝜋
0
−𝜋 ≤ 𝑥 < −2
c. 𝑓(𝑥) =
𝜋
1
−2 ≤𝑥 <
𝜋
2
𝜋
2
𝑓(𝑥 + 2𝜋) = 𝑓(𝑥)
≤𝑥<𝜋
{0
2
d. 𝑓(𝑥) = 1 − 𝑥 , −1 < 𝑥 < 1
0
0≤𝑥<1
e. 𝑓(𝑥) = {
𝑥−1
1≤𝑥<2
0
0≤𝑥<1
f. 𝑓(𝑥) = {
1
1≤𝑥<2
g. 𝑓(𝑥) = 4 − 𝑥 2 , 0 < 𝑥 < 1
𝑥
0≤𝑥<1
h. 𝑓(𝑥) = {
𝑎𝑠 𝑎 𝑠𝑖𝑛𝑒 𝑠𝑒𝑟𝑖𝑒𝑠
1
1≤𝑥<2
0
0≤𝑥<𝜋
i. 𝑓(𝑥) = { 1
𝑎𝑠 𝑎 𝑐𝑜𝑠𝑖𝑛𝑒 𝑠𝑒𝑟𝑖𝑒𝑠
𝜋 ≤ 𝑥 < 2𝜋
2
2𝜋 ≤ 𝑥 < 3𝜋
8. Determine whether the function is even, odd or neither. Based on that analysis, will the Fourier
series modeling the equation in the interval -1<x<1 be a sine series, a cosine series or may
contain terms of both?
a. f ( x )  4 x 3  x 
2
x
2
3
b. g ( x )  2 x  1
x
x 1
|x|
d. j ( x ) 
4x 1
c. h ( x ) 
2
e. k ( x )  4 x 4  x 2  9
f. m( x )  sec x tan x
g. n( x )  sin x  cos x
h. p( x)  e x
2
x 1
i. q( x )  e
j. r(x)=cosh(x)
k. s(x)=sinh(x)