APM2616
Assignment 3 2024 -
DUE 22 July 2024
QUESTIONS WITH DETAILED ANSWERS
[School]
[Course title]
, APM2616 Assignment 3 2024 - DUE 22 July 2024
Question 1:
10 Marks Determine the solution y(x) for each of the following initial value
problems
(1.1) (5) y 0 − yx cos x = 0 with y 0 (π) = 1.
(1.2) (5) 2y 0 + y x = 0, with y 0 (1) = π.
Question 2:
15 Marks Use MuPAD to determine the following indefinite integrals
(2.1) (5) Z x q (2ax − x 2 ) 3 dx.
(2.2) (5) Z 1 x √ 1 + x 2 dx.
(2.3) Obtain a numerical estimate for the following definite integral (5) Z π 2
− π 2 sin x √ 1 + sin x dx 7
Question 3: 25 Marks
The Chebyshev polynomials are defined recursively by the following
formulae: T0(x) = 1, T1(x) = x, Tk (x) = 2x Tk−1(x) − Tk−2(x) for n ≥ 0.
Using MuPad,
(3.1) Write a procedure T(n, t) that takes two inputs n and t and outputs the
Chebychev (15) polynomials.
(3.2) Compute the values of T2(x), ... , T5(x). (5)
(3.3) Compute the values of T2(x), ... , T15(x) for x = 1. (5)
Question 1
1.1. Solve the initial value problem y′−yxcosx=0y' - yx \cos x =
0y′−yxcosx=0 with y(π)=1y(\pi) = 1y(π)=1.
To solve this differential equation, we recognize that it is a first-order linear
differential equation of the form y′+P(x)y=Q(x)y' + P(x)y =
Q(x)y′+P(x)y=Q(x).
Assignment 3 2024 -
DUE 22 July 2024
QUESTIONS WITH DETAILED ANSWERS
[School]
[Course title]
, APM2616 Assignment 3 2024 - DUE 22 July 2024
Question 1:
10 Marks Determine the solution y(x) for each of the following initial value
problems
(1.1) (5) y 0 − yx cos x = 0 with y 0 (π) = 1.
(1.2) (5) 2y 0 + y x = 0, with y 0 (1) = π.
Question 2:
15 Marks Use MuPAD to determine the following indefinite integrals
(2.1) (5) Z x q (2ax − x 2 ) 3 dx.
(2.2) (5) Z 1 x √ 1 + x 2 dx.
(2.3) Obtain a numerical estimate for the following definite integral (5) Z π 2
− π 2 sin x √ 1 + sin x dx 7
Question 3: 25 Marks
The Chebyshev polynomials are defined recursively by the following
formulae: T0(x) = 1, T1(x) = x, Tk (x) = 2x Tk−1(x) − Tk−2(x) for n ≥ 0.
Using MuPad,
(3.1) Write a procedure T(n, t) that takes two inputs n and t and outputs the
Chebychev (15) polynomials.
(3.2) Compute the values of T2(x), ... , T5(x). (5)
(3.3) Compute the values of T2(x), ... , T15(x) for x = 1. (5)
Question 1
1.1. Solve the initial value problem y′−yxcosx=0y' - yx \cos x =
0y′−yxcosx=0 with y(π)=1y(\pi) = 1y(π)=1.
To solve this differential equation, we recognize that it is a first-order linear
differential equation of the form y′+P(x)y=Q(x)y' + P(x)y =
Q(x)y′+P(x)y=Q(x).
