APM2616 Assignment 3 2024 - DUE 22 July 2024

APM2616
Assignment 3 2024
- DUE 22 July 2024
QUESTIONS WITH COMPLETE ANSWERS




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, 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: Determine the solution \( y(x) \) for each of the following initial
value problems


#### (1.1) \( y' - yx \cos x = 0 \) with \( y(\pi) = 1 \).


This is a first-order linear differential equation. To solve it, we use the integrating
factor method.