A3 Find a series expansion, up to x6, for the general solution of Hermite's equation:y" - xy + sy = 0.Write the expansions for the specific solutions y₁, y2, satisfyingy₁ (0) = 1,; y₁ (0) = 0 and y2 (0) = 0, ; 3₂(0) = 1.Use the recurrence relation to show that the series for either y₁ or for y2 truncatesto a polynomial if s is a positive integer.Write down the explicit forms of the expansions for y₁ and y2 for each of the casess 3 and s = 4.

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Answered: Find a series expansion, up to xº, for…

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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Find a general solution in powers of x of the differential equation. State the recurrence relation and the guaranteed radius of convergence. (x²-9) y' + 4xy' + 2y = 0 The recurrence relation is C₁ +2 Find a general solution in powers of x. y(x) = (Type an expression in terms of co and c₁. Type any series in summation notation using n as the index variable and 0 as the starting index.) Find the guaranteed radius of convergence p. p= Find a general solution in powers of x of the differential equation. State the recurrence relation and the guaranteed radius of convergence. (x²+25) y" +6xy' + 4y = 0 The recurrence relation is Cn +2 = Find a general solution in powers of x. y(x) = (Type an expression in terms of co and c₁. Type any series in summation notation using n as the index variable and 0 as the starting index.) Find the guaranteed radius of convergence p. p=
Find the Frobenius series solutions of 2x²y" + 3xy'- (x² + 1)y = 0. %3D
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) =+ ..
Assume the solution of = 5 – y, y(0) = 2 can be written in the form of a power series, y = E-o Cnx". dt Determine the first three nonzero coefficients co, C1, and c2. Then you have a second-degree polynomial that approximates the true solution of the initial value problem. Use your second-degree polynomial to estimate У(1).
Consider a series solutions about the point x = 0 for the ODE: y" - xy + xy = 0 (i) Find a recurrence relationship for the coefficients of the series. (ii) Using (i) to evaluate the first four non-zero terms of each of the two linearly independent solutions and state the general solution for y(x).
The Legendre’s differential equation is (1 − x2)y'' − 2xy' + l(l + 1)y = 0, where l is a constant, and y = y(x). Consider a series solution about the point x = 0 of the form provided, where k is a constant and the an are coefficients that need to be determined. Show that the recurrence relation is given by an+2 (n + k + 2)(n + k + 1) = an [(n + k)(n + k + 1) − l(l + 1)].
SHOW ALL THE STEPS
5) Find the power series solution of the ODE given by y" – xy = 0 in the form y =ECx where m0 y = y(x).
Suppose that f(x) and g(x) are given by the power series and g(x) = 16 + 40x +37x² +48x³ + By dividing power series, find the first few terms of the series for the quotient g(x) f(x) Co = C1 = C₂ = C3 = → f(x) = 4+7x + 2x² + 3x³ + · ← h(x) = = = co + C₁x + c₂x² + 3x³ +
Find the recurrence relation of the coefficients for the two linearly independent power series solutions for the differential equation: y" + 2xy' + 4y = 0
Look for a power series solution of xy" + y' + 2y = 0 centered at x = 1. Find the recurrence relation. You won't be able to find a nice form for the sum, but find up to the (x − 1)²2 terms of each of the solutions y₁ and y2.
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.

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