3. (6.3.1 #6) Find the power series in x for the general solution.(1-x²) y" - 8xy 12y = 0

We have to find the power series solution of

Answered: 3. (6.3.1 #6) Find the power series in…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question 5
d²y + (1-x2) dx2 9. If y = In (cos-1x), show that (1 -x2). dy = x dx dx Hence, find the Maclaurin series for In (cos- x) up to and including the term in x.
4. It can be shown using material from a later section that Set ² d da is given by (−1)na²n+1 n!2n (2n + 1) n=0 for any positive value of a. When a is positive, this series is an alternating series. (a) Approximate dx to four decimal places accuracy by using the Alternating Series Test and its remainder theorem. (Be careful to not round intermediate steps.) (b) Using the same number of terms as you used in your sum in part (a), compute the midpoint approximation Mn for the integral fe-t³² da (c) Which approximation is more accurate? This should be determined using just parts (a) and (b).
I need help with number 2.
+00 (-1)"2n (2n)! 1. For all x € R, cos a = n=0 (a) Find a power series that is equal to x cos(x2) for all a E R. (b) Differentiate the series in (la) to find a power series that is equal to cos(x2) – 2x2 sin(x2) for all a eR. (c) Use the result in (1b) to prove that +oo (-16)" (4n + 1) (2n)! = cos(4) – 8 sin(4). n=0
I need the answer as soon as possible
(c) cOS(7)e " (HINT: write the series as the sum of two series, one of which has only non-negative terms and the second only has negative terms.)
3. Find the Laurent's Series for 3 1/2 f(z) = sinz (Z-Tr) 4 2 f(Z) = z e
(-1)"x²n (2n)! 1. For all x € R, cos ax = n=0 (a) Find a power series that is equal to x cos(x²) for all x E R. (b) Differentiate the series in (la) to find a power series that is equal to cos(x²) – 2x² sin(x²) for all x E R. (c) Use the result in (1b) to prove that +oo Σ (-16)"(4n+ 1) (2n)! cos(4) – 8 sin(4). n=0 W!
Find the sum of the series. (-1)"12" Σ 32n(2n)! n = 0 Step 1 x2n We know that cos(x) = ) (-1)" n = 0 Submit Skip (you cannot come back) 8

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3. (6.3.1 #6) Find the power series in x for the general solution. (1 − x²)y" – 8xy' – 12y = 0