༡ 6. Solve the differential equation by means of a power series about the given ordinary point .xo. Find the recurrence relation for the coefficients of the power series, and also write out the first four terms in each of the two fundamental solutions yı (t) and y2(t). (a) y" +2y=0, x0 = 0 (b) y" + xy+2y= 0, x0 = 0 (c) xy"+y+xy = 0, x0 = 1
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- A well known equation in mathematical physics is the equation y" - 2xy + 2ay = 0 n=0 Find the power series solution, y = no anx". Write your answer in terms of the even and odd series y(x) = Coy₁ (x) + C1y2(x), respectively. Find three (3) terms in each series solution, y₁ (x) and y₂(x). Find the polynomial solutions for a = 0, 1, 2, 3, 4 What do you notice?Consider the Chebyshev di↵erential equation(1 x2)y00 xy0 + 2y = 0 ,where is a constant. Find the power series solutions of this di↵erential equation aboutx0 = 0. For what values of the constant these solutions reduce to polynomials, known asChebyshev polynomials. Compute first four of the Chebyshev polynomialssolve the differential equation (3-x2)y''-3xy'-y=0 by means of power series about the ordinary point x=0 .Find the recurrence relation; also find the first four terms in each of two linearly independent solutions. If possible, find thegeneral term in each solution.
- 1. (50) Solve the differential equation (3 – 2)y" – 3xy' – y = 0 by means of a power series about the ordinary point x = 0. Find the recurrence relation; also find the first four terms in each of two linearly independent solutions. If possible, find the general term in each solution.This is a differential equations problem. Seek a power series solution of the equation y'' +2xy' +2y = 0 about the point x=0. a. Find the recurrence relation. b. Find the first 3 nonzero terms in each of two solution y1 and y2 which form the fundamental set of solutions.Solve using a Power Series Centered at 0. Express your answer in the form y = c0(First three non-zero terms of Series1+ · · ·)+c1(First 3 non-zero terms of Series2+ · · ·). You do not have to simplify/multiply the coefficients y'' + x2y' = 0
- 1). Solve the differential equation 3y" - 3x2y" - 6xy' + 18y = 0 . Then evaluate the first three terms of the infinite series at x = 0.755 2). Solve the differential equation y" + 12y = x2y"+ 2xy' . Then evaluate the Legendre polynomial at x = 0.673Find the first five nonzero terms of a power series solution about t=0 for the ODE y''+ty=0. Simplify the coefficients completely.olve the given differential equation by means of a power series about the given point x0. Find the recurrence relation; also find the first four terms in each of two linearly independent solutions (unless the series terminates sooner). If possible, find the general term in each solution. I did not how this question was solved like this. I do not understand why when a(n+3)=-nan/(n+2)(n+1) a3=a4=a5=.....=0. Please help me go over this question. thank you. The origin equation is y''+xy'=0 x0=0
- Find the recursion formula for the coefficients of the power series. For n = 10, what is the factor between (an) + 2 and (an) (meaning, what is (an) + 2/(an))?x²y" + xy' + (x − 2) y = 0Let f(x) = 1 + x 1 X Find the power series representation for the function f(x) by completing the following steps: a. First, express the fraction 1¹ as a power series. = X b. Now, express the fraction as a power series. 1-x 1+x x c. The function f(x) 1-x 1-x + 1 is the sum of the two series from parts (a) and (b). Express the function f(x) as a power series. d. What is the interval of convergence and the radius of convergence for this power series?