Math 285 Final Exam supplement Study Guide
Summer 2012
5.1. Review of Power Series & Solutions Near an Ordinary Point
 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.
 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.
6.2 Solutions around Singular Points
 Know the difference between an ordinary point and a singular point
 Know the difference between a regular singular point and an irregular singular point.
 Be able to solve for solutions near a regular singular point using Frobenius’ Theorem.
7.1 Laplace Transforms
 Know the definition of a Laplace transform and be able to solve for the F(s) function through
integration.
 Know what it means to be of exponential order
 Be able to find the transform of a function using a table of common functions
7.2 Inverse Laplace Transforms and Derivatives
 Be able to find inverse Laplace transforms using a table of common functions.
 Review Partial Fractions
 Be able to find the Laplace transform of the derivative of a function.
 Be able to use initial values to solve differential equations with Laplace transforms.
7.3 Operational Properties I
 Know the first translation theorem and how to use it.
 Know how to write a piecewise function in terms of the unit step function
 Know the two versions of the second translation theorem
 Be able to use the translation theorems to solve initial value problems
7.4 Operational Properties II
 Know how derivatives of transforms relate to original functions
 Know what a convolution is and how to transform it.
 Know how to transform an integral of a function.
 Be able to state the transform of a periodic function
 Be able to solve an integrodifferential equation (one involving both derivatives and integrals)
using Laplace transforms.
Plus: review major topics from previous exams!
Problems:
1. Determine the interval and radius of convergence.
( 1)n 24 n x n

n 0 (2n  1)!

a.
( x  1)n
n!
n 1

b.


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 ordinary point. 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. Use Frobenius’ Theorem to solve the differential equation (or state that it is impossible).
Classify each singular point as regular or irregular. (Assume the point of expansion is 𝑥0 = 0
unless stated otherwise.
a. 𝑥(𝑥 + 3)2 𝑦 ′′ − 𝑦 = 0
1
1
b. 𝑦 ′′ − 𝑦 ′ +
3𝑦 = 0
𝑥
(𝑥−1)
c. 2𝑥𝑦 ′′ − 𝑦 ′ + 2𝑦 = 0
d. 2𝑥 2 𝑦 ′′ − 𝑥𝑦 ′ + (𝑥 2 + 1)𝑦 = 0
e. 𝑥𝑦 ′′ = 𝑥𝑦 ′ + 𝑦 = 0
6. Find the Laplace transform of the following functions using the definition (a-f only, use the table
after that). When possible, compare your results to the table of common functions.
−1, 0 ≤ 𝑡 < 1
a. 𝑓(𝑡) = {
1,
𝑡≥1
2𝑡 + 1, 0 ≤ 𝑡 < 1
b. 𝑓(𝑡) = {
0,
𝑡≥1
4𝑡
c. 𝑓(𝑡) = 𝑡𝑒
d. 𝑓(𝑡) = 𝑡 5
e. 𝑓(𝑡) = cos 5𝑡 + sin 2𝑡
f. 𝑓(𝑡) = cosh 4𝑡
g. 𝑓(𝑡) = 𝑒 𝑡 𝑠𝑖𝑛3𝑡
h. 𝑓(𝑡) = 𝑒 2𝑡 (𝑡 − 1)2
i. 𝑓(𝑡) = 𝑡 2 ∗ 𝑡𝑒 𝑡
𝑡
j. 𝑓(𝑡) = ∫0 𝜏 sin 𝜏 𝑑𝜏
𝑡
k. 𝑓(𝑡) = ∫0 𝜏𝑒 𝑡−𝜏 𝑑𝜏
7. Find the inverse Laplace transform of the following functions.
1
a. 𝐹(𝑠) = 2
𝑠
1
b. 𝐹(𝑠) = 4𝑠+1
5
c. 𝐹(𝑠) = 𝑠2 +49
d. 𝐹(𝑠) =
e. 𝐹(𝑠) =
f.
𝐹(𝑠) =
g. 𝐹(𝑠) =
h. 𝐹(𝑠) =
i.
𝐹(𝑠) =
2𝑠−6
𝑠2 +9
𝑠
𝑠2 +2𝑠−3
2𝑠−4
(𝑠2 +𝑠)(𝑠2 +1)
1
(hint:
𝑠2 +2𝑠+5
2
(𝑠+1)
(𝑠+2)4
1
𝑠(𝑠−1)
complete the square)
8. Use Laplace Transforms to solve the given initial value problem. Check your answer by using
another method.
a. 𝑦 ′ − 𝑦 = 1, 𝑦(0) = 0
b. 𝑦 ′ + 6𝑦 = 𝑒 4𝑡 , 𝑦(0) = 2
c. 𝑦 ′′ + 5𝑦 ′ + 4𝑦 = 0, 𝑦(0) = 1, 𝑦 ′ (0) = 0
d. 𝑦 ′′ − 4𝑦 ′ = 6𝑒 3𝑡 − 3𝑒 −𝑡 , 𝑦(0) = 1, 𝑦 ′ (0) = −1
e. 𝑦 ′′ − 4𝑦 ′ + 4𝑦 = 𝑡 3 𝑒 2𝑡 , 𝑦(0) = 0, 𝑦 ′ (0) = 0
f. 𝑦 ′′ − 𝑦 ′ = 𝑒 𝑡 cos 𝑡 , 𝑦(0) = 0, 𝑦 ′ (0) = 0
𝑡, 0 ≤ 𝑡 < 1
g. 𝑦 ′ + 2𝑦 = 𝑓(𝑡), 𝑦(0) = 0, 𝑓(𝑡) = {
0,
𝑡≥1
h. 𝑦 ′′ + 4𝑦 = sin 𝑡 𝒰(𝑡 − 2𝜋), 𝑦(0) = 1, 𝑦 ′ (0) = 0
𝑡
i. 𝑦 ′ (𝑡) = 1 − sin 𝑡 − ∫0 𝑦(𝜏)𝑑𝜏 , 𝑦(0) = 0