APM2611
ASSIGNMENT 4 2024
UNIQUE NO.
DUE DATE: 25 SEPTEMBER 2024
, APM2611/101/0/2024
ASSIGNMENT 04
Due date: Wednesday, 25 September 2024
-
ONLY FOR YEAR MODULE
Series solutions, Laplace transforms and Fourier series, solving PDE’s by separation of
variables.
Answer all the questions. Show all your own and personalized workings, you get ZERO
to a question if we see that you have copied someone’s else solution word by word.
If you choose to submit via myUnisa, note that only PDF files will be accepted.
Note that all the questions will be marked therefore, it is highly recommended to attempt all of them.
Question 1
1. Find the radius and interval of convergence of the following series:
X∞ (−1) n−1 x 2n−1
n=1
(2n − 1)!
2. Rewrite the expression below as a single power series:
X∞ X∞
cn+1 x n−2 − 4cn x n−1 .
n=2 n=1
3. Use the power series method to solve the initialvalue problem
00 0
(x + 1)y − (2 − x)y + y = 0, y(0) = 2, y0(0) = −1;
where c0 and c1 are given by the initial conditions.
4. Use the power series methodPto solve the initialvalue problem.In particular, find c 0, c1 , c2, c3
and c4 in the equation y(x) = ∞n=0 cn x n .
y00− x 2y = 0; y(0) = 3, y0(0) = 7.
Question 2
Using the method of separation of variables, find a solution for the following PDEs:
1.
∂ 2u ∂u
=4 .
∂x 2 ∂y
19
, 2.
∂u ∂u
= .
∂x ∂y
3. the initial value problem:
y00+ 4y 0 + 3y = 1 − U(y − 2) − U(t − 4) + U(t − 6),
y(0) = 0, y0(0) = 0.
Question 3
Compute the following Fourier series:
1. Fourier series for f (x) = x 2
on [−π, π].
2. Sine series for f (x) = x 2 on [0, π].
3. Cosine series for f (x) = x 2 on [0, π].
Question 4
Using the method of separation of variables, find a solution for the following PDEs:
∂ 2u ∂u
1. = .
∂x 2 ∂y
∂u ∂u
2. =4 .
∂x ∂y
Question 5
Find the temperature u(x, t) in a rod of length L if the initial temperature is f (x) throughout and if
the ends x = 0 and x = L are insulated.
– End of assignment –
20
ASSIGNMENT 4 2024
UNIQUE NO.
DUE DATE: 25 SEPTEMBER 2024
, APM2611/101/0/2024
ASSIGNMENT 04
Due date: Wednesday, 25 September 2024
-
ONLY FOR YEAR MODULE
Series solutions, Laplace transforms and Fourier series, solving PDE’s by separation of
variables.
Answer all the questions. Show all your own and personalized workings, you get ZERO
to a question if we see that you have copied someone’s else solution word by word.
If you choose to submit via myUnisa, note that only PDF files will be accepted.
Note that all the questions will be marked therefore, it is highly recommended to attempt all of them.
Question 1
1. Find the radius and interval of convergence of the following series:
X∞ (−1) n−1 x 2n−1
n=1
(2n − 1)!
2. Rewrite the expression below as a single power series:
X∞ X∞
cn+1 x n−2 − 4cn x n−1 .
n=2 n=1
3. Use the power series method to solve the initialvalue problem
00 0
(x + 1)y − (2 − x)y + y = 0, y(0) = 2, y0(0) = −1;
where c0 and c1 are given by the initial conditions.
4. Use the power series methodPto solve the initialvalue problem.In particular, find c 0, c1 , c2, c3
and c4 in the equation y(x) = ∞n=0 cn x n .
y00− x 2y = 0; y(0) = 3, y0(0) = 7.
Question 2
Using the method of separation of variables, find a solution for the following PDEs:
1.
∂ 2u ∂u
=4 .
∂x 2 ∂y
19
, 2.
∂u ∂u
= .
∂x ∂y
3. the initial value problem:
y00+ 4y 0 + 3y = 1 − U(y − 2) − U(t − 4) + U(t − 6),
y(0) = 0, y0(0) = 0.
Question 3
Compute the following Fourier series:
1. Fourier series for f (x) = x 2
on [−π, π].
2. Sine series for f (x) = x 2 on [0, π].
3. Cosine series for f (x) = x 2 on [0, π].
Question 4
Using the method of separation of variables, find a solution for the following PDEs:
∂ 2u ∂u
1. = .
∂x 2 ∂y
∂u ∂u
2. =4 .
∂x ∂y
Question 5
Find the temperature u(x, t) in a rod of length L if the initial temperature is f (x) throughout and if
the ends x = 0 and x = L are insulated.
– End of assignment –
20
