Show that x = 0 is a regular singular point for the equation (Bessel's equation ofzero order)xy" + y' + xy = 0and find the corresponding indicial equation.Find a series solution of the formy = 1 + Σ anx",n=1deriving the general recurrence relation and calculating the explicit form of an.= 0, k = 1, 2, … ... Determine first four nonzero terms of theShow that a2k-1 =expansion.

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Answered: Show that x = 0 is a regular singular…

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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4,8) Please type answer or make it very easy to read
3) Find the first five terms, ao, a₁, a2, a3, a4, in the series solution at x = 0 of the initial value problem (IVP) (x - 1)²y" + xy + y = 0, y(0) = 0, y'(0) = 1.
4
Find the first three nonzero terms, or as many as exist, in the series expansion about x= 0 for a general solution to the given equation for x>0. ху" + (1- х)у' - у %3D0 The equation has a general solution in the form C, y, (x) + cay2(x) where y, (x) is obtained from the root of the associated indicial equation with the largest real value. If the second solution has term with y, (x), then count that term as only one term. y1(X) =+ .. and y2(x) =+ ..
Q3) a) Evaluate fx-²1₂(3x) dx. b) Expand the following function in a Fourier Bessel series using Bessel functions of the same order as in the indicated boundary condition: ƒ(x)=x², 0
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2) Suppose you have a differential equation that satisfies the conditions of Theorem 6.2.1- Existence of Power Series Solutions with center x, = 0. It is then guaranteed that a solution exists of the form y(x) = E-0 Cnx". The recurrence relation for the coefficients was calculated to be k 3 Ck+2 k = 1,2,3, .. 2(k + 2) k+1 2(k + 2)(k + 1) 3 C2 = -7C0 Find the solution y(x) to the initial-value problem if y(0) = 0 and y'(0) = 1. Give at least the first 3 nonzero terms of the solution.
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1.) Given xy^''+2y^'-y=0 solve by the method of Frobenius when r=1 and conclude whether r=1 is correct value or can be used to solve the differential equation. After identifying the first three terms, which will become the leader of the REMAINING SERIES? Show how the 2 other series will become a member of the REMAINING SERIES Calculate the RECURRENCE FORMULA: the values of a3, a4, & a5 Conclusion: Is the proper value obtained from the indicial equation? Yes or no, briefly explain.
2- Consider the differential equation (²+3)y" - 4y' = 0. The recurrence relation for the coefficients an of the power series solution an" about the ordinary point o=0 is given by b) an+2= an+2= an+2= none of these an+2= 12-0 4nan+1 + (n² = n)an 3n² +9n+6 n22 (4n+4)an+1-(n² - n)an 3n² +9n+6 (n + 4)an+1 +n²an 3n² +9n+6 an+1 + (n²-n)an 3n² +9n+6 1 " n>2 n≥ 2 n22
1) e(x- 2)" student solution for (T.S): n=0 2" n! a) e?x b) et c) e? d) sin 2x 2)What are the first four terms of the Taylor series for e* when a = 1? e- =e+e(x-1) + 늘c(-1)2 A. +e(x- 1)3 +... e-= = e – e(x – 1) + e(x – 1)? B. -e(x – 1)3 +... e = !+(- 1) + (z – 1)2 c + (= – 1)° + .. c= = !- (x – 1) + (x – 1)? D- (E – 1)* + ... | 3)What is the x term in the Taylor expansion for xe* when a = 0? A) B) C)F DY 4) What is the fourth term in the Taylor expansion for 1 when a = 1 ? A)2(1 – 2) B)2(z – 1)° C)(1- z) D )(7- 1)
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