Exam (elaborations)
APM2616 Assignment 3 2024 - DUE 22 July 2024
- Course
- Computer Algebra (APM2616)
- Institution
- University Of South Africa
APM2616 Assignment 3 2024 - DUE 22 July 2024 QUESTIONS AND ANSWERS
[Show more]
Content preview
[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′−yxcosx=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)=−xcosxP(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:
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through EFT, credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying this summary from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller inotes. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy this summary for R45,43. You're not tied to anything after your purchase.
Can Stuvia be trusted?
4.6 stars on Google & Trustpilot (+1000 reviews)
85443 documents were sold in the last 30 days
Founded in 2010, the go-to place to buy summaries for 14 years now