APM2616 Assignment 3 2024 - DUE 22 July 2024

[Date]




APM2616 Assignment
3 2024 - DUE 22 July
2024
QUESTIONS AND ANSWERS

, 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′−yxcos⁡x=0y' - yx \cos x = 0y′−yxcosx=0
with y(π)=1y(\pi) = 1y(π)=1.
To solve this first-order linear differential equation, we can use the integrating
factor method. The general form of a first-order linear differential equation is:
y′+P(x)y=Q(x)y' + P(x)y = Q(x)y′+P(x)y=Q(x)
Here, P(x)=−xcos⁡xP(x) = -x \cos xP(x)=−xcosx and Q(x)=0Q(x) = 0Q(x)=0.
The integrating factor μ(x)\mu(x)μ(x) is given by: