Logo
Home Study Material Contact Search
AI Chat FREE
  1. business
  2. accounting
  3. section 5.1 reference: please neatly show all wor

SECTION 5.1 REFERENCE: Please neatly show all work. Thank you! Problem 1. Does...

90.2K

Verified Solution

Question

Accounting

image

SECTION 5.1 REFERENCE:

image

Please neatly show all work. Thank you!

Problem 1. Does this ODE x+y" + (3x 1)y' + y = 0 have such a power series solution y = En anxwith radius of convergence R > 0? If so, please find it; if not, please prove your judgement and explain why this does not contradict the main theorem in Section 5.1. (Hint: Recall Cauchy - Hadamard formula: 1 R lim n-00 an+1 an provided the limit exists (can be infinity).) When do power series solutions exist? Answer: if p, q, r in the ODES (12) y + 2xy + g(x)) = (x) have power series representations (Taylor series). More precisely, a function f(x) is called analytic at a point x = xo if it can be represented by a power series in powers of x Xo with positive radius of convergence. Using this concept, we can state the following basic theorem, in which the ODE (12) is in standard form, that is, it begins with the y". If your ODE begins with, say, h(x))", divide it first by h(x) and then apply the theorem to the resulting new ODE. THEOREM 1 Existence of Power Series Solutions If p, q, and r in (12) are analytic at x = Xo, then every solution of (12) is analytic at x = Xo and can thus be represented by a power series in powers of x Xo with radius of convergence R>0

Answer & Explanation Solved by verified expert
Get Answers to Unlimited Questions

Join us to gain access to millions of questions and expert answers. Enjoy exclusive benefits tailored just for you!

Membership Benefits:
  • Unlimited Question Access with detailed Answers
  • Zin AI - 3 Million Words
  • 10 Dall-E 3 Images
  • 20 Plot Generations
  • Conversation with Dialogue Memory
  • No Ads, Ever!
  • Access to Our Best AI Platform: Flex AI - Your personal assistant for all your inquiries!
Become a Member

Other questions asked by students