Differential Equation Assignment

Problem 3. Consider the following initial value problem, y 00 + y = g(x), y(x0) = 0, y0 (x0) = 0, x ∈ [0, ∞).

a. Show that the general solution of y 00 + y = g(x) is given by y = φ(x) =  c1 − Z x α g(t) sin t dt  cos x +  c2 + Z x β g(t) cost dt  sin x, where c1, c2 are arbitrary constants and α, β are any conveniently chosen points.

b. Using the result of (a) show that y(x0) = 0 and y 0 (x0) = 0 if, c1 = Z x0 α g(t) sin t dt, c2 = − Z x0 β g(t) cost dt, 1 and hence the solution of the above initial value problem for arbitrary g(x) is, y = φ(x) = Z x x0 g(t) sin(x − t) dt. Notice that this equation gives a formula for computing the solution of the original initial value problem for any given nonhomogeneous term g(x). The function φ(x) will not only satisfy the differential equation but will also automatically satisfy the initial conditions. If we think of x as time, the formula also shows the relation between the input g(x) and the output φ(x). Further, we see that the output at time x depends only on the behavior of the input from the initial time x0 to the time of interest. This integral is often referred to as the convolution of sin x and g(x).

c. Now that we have the solution of the linear nonhomogeneous differential equation satisfying homogeneous initial conditions, we can solve the same problem with nonhomogeneous initial conditions by superimposing a solution of the homogeneous equations satisfying nonhomogeneous initial conditions. Show that the solution of y 00 + y = g(x), y(x0 = 0) = y0, y0 (x0 = 0) = y 0 0 , is y = φ(x) = Z x x0 g(t) sin(x − t) dt + y0 cos x + y 0 0 sin x.

Problem 4. The Tchebycheff (1821-1894) differential equation is (1 − x 2 )y 00 − xy0 + α 2 y = 0, α constant.

a. Determine two linearly independent solutions in powers of x for |x| < 1. For α = 1 graph 5 and 10 terms of both solutions.

b. Resolve this problem using (1) the dsolve command, and (2) the dsolve command with “series” option in MAPLE (or equivalent commands in Mathematica or Matlab). Plot these results on your curves obtained in part a.

c. Show that if α is a non-negative integer n, then there is a polynomial solution of degree n. These polynomials, when properly normalized, are called Tchebycheff polynomials. They are very useful in problems requiring a polynomial approximation to a function defined on −1 ≤ x ≤ 1.

d. Find polynomial solutions for each of the cases n = 0, 1, 2, 3.

 

We confidently assure you high-quality work. We engage a number of strategies in to guarantee top-level assignments. The projects go through a thorough system of control prior to being submitted to the customers. Studentscoursework.com guarantees that the assignments delivered are 100 percent plagiarism-free.

We never share or resell customer’s work. Therefore, once it is checked using our plagiarism software, the assignment will be original. In terms of assignment completion, we adhere to the guidelines given by you and assign the work to the most suitable writer. We assign our writers based on various factors such as academic qualification, previous customer feedback, etc.

First and foremost, we provide high-quality work. However, in case you require extra-brilliance, here is what you can do. First, select a top writer who is proficient in a certain discipline. Second, you can seek our editing services.

Our team of editors revises the assignments, checking them to ensure they comply with the standards of academic writings. Additionally, editing may entail refining the language, adding more reference material, and making sure the formatting part is properly done. The editors may also enhance papers completed by yourself in to meet your needs.

We guarantee the following quality in our work to you:

  • Properly cited scholarly sources from assigned or outside reading and research.
  • No wordy, vague or poorly constructed sentences
  • Clearly written and structurally sound sentences, with no grammar, spelling and/or punctuation errors.
  • Assignments with proper syntax, that demonstrates mastery of the subject, and with clear organization flow.
  • Strict adherence to requested citation styles (i.e. APA, MLA, HARVARD, CHICAGO/TURABIAN)
  • Proper page numbering, heading, spacing and margins.
  • Parenthetical citations are in the proper format, are properly punctuated and are used at appropriate points to document source material.
  • Plagiarism free guaranteed work.
  • Expert proofreading.

We use the following tools to crosscheck our assignments:

  • Turnitin
  • Grammarly

 

Place your order
(550 words)

Approximate price: $22

Calculate the price of your order

550 words
We'll send you the first draft for approval by September 11, 2018 at 10:52 AM
Total price:
$26
The price is based on these factors:
Academic level
Number of pages
Urgency
Basic features
  • Free title page and bibliography
  • Unlimited revisions
  • Plagiarism-free guarantee
  • Money-back guarantee
  • 24/7 support
On-demand options
  • Writer’s samples
  • Part-by-part delivery
  • Overnight delivery
  • Copies of used sources
  • Expert Proofreading
Paper format
  • 275 words per page
  • 12 pt Arial/Times New Roman
  • Double line spacing
  • Any citation style (APA, MLA, Chicago/Turabian, Harvard)

Our guarantees

Delivering a high-quality product at a reasonable price is not enough anymore.
That’s why we have developed 5 beneficial guarantees that will make your experience with our service enjoyable, easy, and safe.

Money-back guarantee

You have to be 100% sure of the quality of your product to give a money-back guarantee. This describes us perfectly. Make sure that this guarantee is totally transparent.

Read more

Zero-plagiarism guarantee

Each paper is composed from scratch, according to your instructions. It is then checked by our plagiarism-detection software. There is no gap where plagiarism could squeeze in.

Read more

Free-revision policy

Thanks to our free revisions, there is no way for you to be unsatisfied. We will work on your paper until you are completely happy with the result.

Read more

Privacy policy

Your email is safe, as we store it according to international data protection rules. Your bank details are secure, as we use only reliable payment systems.

Read more

Fair-cooperation guarantee

By sending us your money, you buy the service we provide. Check out our terms and conditions if you prefer business talks to be laid out in official language.

Read more

Order your paper today and save 30% with the discount code HAPPY

X
Open chat
1
You can contact our live agent via WhatsApp! Via + 1 323 412 5597

Feel free to ask questions, clarifications, or discounts available when placing an order.

Order your essay today and save 30% with the discount code HAPPY