HOMEWORK 4 FOR M427K-FALL 2014

| August 31, 2017

Question
HOMEWORK 4 FOR M427K-FALL 2014

1. Notes
Homework 4 consists of 4 problems. The total grade for the first three problems is 60 points,
which is also the total grade for Homework 4. Problem 4 below is the bonus point problem,
and is worth 20 points.
For all questions regarding the grading of Homework 4, please ask the TA.
Please submit your homework to the TA, Juan Diego, before the problem session on Tuesday,
November 18 starts. Late homework will not be accepted.
2. Problems in Homework 4
Please solve the following problems in full detail.
Problem 1. (20 points) Compute the inverse Laplace transform of the function F (s) given by
F (s) =

s3 + 2s2 + 2s + 1
.
(s − 3)5

That is, compute L−1 {F (s)}.
Problem 2. (20 points) Use the Laplace transform to solve the following initial value problem
y − 7y + 6y = 5t,

y(0) = 1, y (0) = 3.

Problem 3. (20 points) Use the Laplace transform to solve the following initial value problem
y + 2y − 3y = te−3t ,

y(0) = 0, y (0) = 5.

Problem 4. (20 points) Recall that the binomial theorem states that
n

(a + b)n =
k=0

n k n−k
a b
k

for any positive integer n. Here
n
k

=

n!
k!(n − k)!

for each 0 ≤ k ≤ n.
Let n be a positive integer. Use the Laplace transform and the binomial theorem to solve the following
initial value problem:
y

n
n
k=0 k
(n−1)

y (k) = e−t
(0) = y (n−2) (0) = · · · = y (2) (0) = y (0) = y(0) = 0.

Here, for each 0 ≤ k ≤ n, y (k) denotes the k-th derivative of y.

Date: November 13, 2014.
1

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