In each of Problems 1 through 11:a. Seek power series solutions of the given differential equation about the given point xo; find the recurrence relation that the coefficients must satisfy.b. Find the first four nonzero terms in each of two solutions y₁ and y2 (unless the series terminates sooner).c. By evaluating the Wronskian W[y₁,2] (x0), show that y₁ and y2 form a fundamental set of solutions.d. If possible, find the general term in each solution.7.y" + xy + 2y = 0,x0 = 011. 2y" + (x + 1)y' + 3y = 0,ka contx0 = 2

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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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3. Find infinite series solution for the following DE about the regular singular point x=0, a. 2x²y" + 3xy' - (x² + 1)y = 0 b. 2x²y" - 2xy + x² y = 0 c. x²y" −xy' + (x + 1) y = 0
2. Use the Frobenius series to find the solution of in some interval 0 < x < d. 1²y" +1(2+1)y' - 2y = 0
2. Seek a power series solution of the form y(r) =ana" to the differential equation n=0 y"(r) – 2ry (x) – 4y(x) = 0, xo = 0. (a) Set-up a recurrence for the cofficients, an. (b) Find the first four terms of two independent power series solutions yı and y2 in such a way that you can write y(r) = a0y1(x) + a1y2(x). (c) Determine the Taylor series of e about the point ro = 0. %3D (Hint: Use the Taylor series for e".) (d) By using Part(c), find the explicit form of y2(x).
4. Use the table in the book to find the Taylor series for the function, centered at the specified c (or c=0 if none is specified). f(x)=e³x f(x) = ln(x² +1) f(x)=sinhx f(x)=2 sin x³ f(x)=e* cos x a. b. c.
Yt = 0.6y -1 + 0.2yt-2 + €t, where e is iid N(0, o²). 2.1) Write down the characteristic polynomial equation and find the characteristic roots of this polynomial. Is yt a stationary time series process?
1. Write the first four terms in the Taylor series of f (x) centered at x = 2 given the following: f(0) = 1, f(1) = 2, f(2) = 3 f'(0) = 4, f'(1) = 5, f'(2) = 6 f"(0) = 7, f"(5) = 8, f"(2) = 9 f(3) (0) = 10, f(3) (3) = 11, f(3) (2) = 12 f(4) (0) = 13 %3D
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(b) Find the number of terms that are to be retained if an accuracy of 10-10 is required in solving the initial value problem dy = x + y, y (0) =1,xe ]0,1[ dx by Taylors' series.
#9: Consider the following differential equation. (1 + 3x²) y - 8xy + 6y = 0 (a) If you were to look for a power series solution about xo = 0, i.e., of the form Σcnx" n=0 then the recurrence formula for the coefficients would be given by ck+2= g(k) ck, k> 2. Enter the function g(k) into the answer box below. (b) Find the solution to the above differential equation with initial conditions y(0) = 0 and y'(0) = 5. (Note that this solution is a terminating power series.) (c) Find the first three nonzero terms in the solution to the above differential equation with initial conditions y(0) = 7 and y'(0) = 0.
1. Given the function f(x) = −2x³ + 12x²-20x+8,5 Using the Taylor series of order zero, one, two, and three, estimate the function at the point Xi+1= 0,5 based on the function value x;= 0.

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