APM2611
ASSIGNMENT 3 2024
UNIQUE NO.
DUE DATE: 14 AUGUST 2024
, ASSIGNMENT 03
Due date: Wednesday, 14 August 2024
-
ONLY FOR YEAR MODULE
First order separable, linear, Bernoulli, exact and homogeneous equations. Higher order
homogeneous DE’s. Solving non-homogeneous DE’s using the undetermined
coefficients, variation of parameters and operator methods.
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.
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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:
(i)
X∞ 100n n
(x + 7)
n=1
n!
(ii)
X∞ (−1) k k
k (x − 5)
k=1
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2. Rewrite the expression below as a single power series:
X∞ n
X∞ X∞
n(n − 1)c n x + 2 n(n − 1)c n x n−2 + ncn x n .
n=2 n=2 n=1
Question 2
1. Verify by direct substitution that the given power series is a particular solution of the DE
00 0
X∞ (−1) n+1 n
(x + 1)y + y = 0 ; y = x .
n
n=1
2. Use the power series method to solve the initialvalue problem
00 0 0
(x + 1)y − (2 − x)y + y = 0, y(0) = 2, y (0) = −1;
where c0 and c1 are given by the initial conditions.
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, APM2611/101/0/2024
Question 3
Calculate the Laplace transform of the following function from first principles:
1.
sin t if 0≤t<π
f (t) =
0 if t≥π
2. f (t) = e −t sin t
3. Use Theorem 7.1 to find L{f (t)}
(i) f (t) = −4t 2
+ 16t + 9
(ii) f (t) = 4t 2
− 5 sin 3t
(iii) f (t) = (e t − e −t ) 2
Question 4
1. Use Theorem 7.3 to find the inverse transform:
(i)
2s − 4
L −1
(s2 + s)(s 2 + 4)
(ii)
s
L −1
(s + 2)(s 2 + 4)
2. Use the Laplace transform to solve the initialvalue problem
y00+ 5y 0 + 4y = 0, y(0) = 1, y0(0) = 0.
Question 5
1,
1. When g(t) = 1 and L{g(t)} = G(s) = s
the convolution theorem implies that the Laplace
transform of the integral of f is
Z t
F (s)
L f (τ ) dτ = .
0
s
The inverse form is
Z t
F (s)
f (τ ) dτ = L −1 .
0
s
Find
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