To determine The solution of the differential equation y ″ + y = 0 by power series method. Explanation Definition used: Power series: “The general power series is an infinite series of the form, ( 1 ) ∑ m = 0 ∞ a m ( x − x 0 ) m = a 0 + a 1 ( x − x 0 ) + a 2 ( x − x 0 ) 2 + ⋯ . Here, x is a variable. a 0 , a 1 , a 2 , ⋯ are constants, called the coefficients of the series. x 0 is a constant, called the center of the series. In particular, if x 0 = 0 , a power series in powers of x ( 2 ) ∑ m = 0 ∞ a m x m = a 0 + a 1 x + a 2 x 2 + a 3 x 3 + ⋯ .” Calculation: The given ODE is y ″ + y = 0 . Consider the power series, y = a 0 + a 1 x + a 2 x 2 + a 3 x 3 + a 4 x 4 + a 5 x 5 + ⋯ = ∑ m = 0 ∞ a m x m . Differentiate the series with respect to x . y ′ = a 1 + 2 a 2 x + 3 a 3 x 2 + 4 a 4 x 3 + 5 a 5 x 4 + ⋯ = ∑ m = 1 ∞ m a m x m − 1 Again differentiate the series with respect to x . y ″ = 2 a 2 + 2 ⋅ 3 a 3 x + 3 ⋅ 4 a 4 x 2 + 4 ⋅ 5 a 5 x 3 + ⋯ = ∑ m = 2 ∞ m ( m − 1 ) a m x m − 2 Substitute the values of y and y ″ in the given ODE. [ ( 2 a 2 + 2 ⋅ 3 a 3 x + 3 ⋅ 4 a 4 x 2 + 4 ⋅ 5 a 5 x 3 + ⋯ ) + ( a 0 + a 1 x + a 2 x 2 + a 3 x 3 + a 4 x 4 + ⋯ ) ] = 0 ( 2 a 2 + a 0 ) + ( 6 a 3 + a 1 ) x + ( 12 a 4 + a 2 ) x 2 + ( 20 a 5 + a 3 ) x 3 + ⋯ = 0 Equate the coefficients of x , x 2 , x 3 to 0

10th Edition

ISBN: 9781119455929

Author: Kreyszig

Publisher: WILEY

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Chapter 5.1, Problem 9P

To determine

The solution of the differential equation y″+y=0 by power series method.

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Ch. 5.1 - WRITING AND LITERATURE PROJECT. Power Series in...Ch. 5.1 - Determine the radius of convergence. Show the...Ch. 5.1 - Determine the radius of convergence. Show the...Ch. 5.1 - Determine the radius of convergence. Show the...Ch. 5.1 - Determine the radius of convergence. Show the...Ch. 5.1 - Apply the power series method. Do this by hand,...Ch. 5.1 - Apply the power series method. Do this by hand,...Ch. 5.1 - Apply the power series method. Do this by hand,...Ch. 5.1 - Apply the power series method. Do this by hand,...Ch. 5.1 - Find a power series solution in powers of x. Show...

Ch. 5.1 - Find a power series solution in powers of x. Show...Ch. 5.1 - Find a power series solution in powers of x. Show...Ch. 5.1 - Find a power series solution in powers of x. Show...Ch. 5.1 - Find a power series solution in powers of x. Show...Ch. 5.1 - Prob. 15P Ch. 5.1 - Prob. 16P Ch. 5.1 - CAS PROBLEMS. IVPs Solve the initial value problem...Ch. 5.1 - Prob. 18P Ch. 5.1 - Prob. 19P Ch. 5.2 - Legendre functions for n = 0. Show that (6) with n...Ch. 5.2 - Legendre functions for n = 1. Show that (7) with n...Ch. 5.2 - Special n. Derive (11′) from (11). Ch. 5.2 - Prob. 4P Ch. 5.2 - Obtain P6 and P7. Ch. 5.2 - Prob. 11P Ch. 5.2 - Prob. 12P Ch. 5.2 - Rodrigues’s formula. Obtain (11′) from (13). Ch. 5.2 - Prob. 14P Ch. 5.2 - Prob. 15P Ch. 5.3 - Prob. 1P Ch. 5.3 - Prob. 2P Ch. 5.3 - Prob. 3P Ch. 5.3 - Prob. 4P Ch. 5.3 - Prob. 5P Ch. 5.3 - Prob. 6P Ch. 5.3 - Prob. 7P Ch. 5.3 - Prob. 8P Ch. 5.3 - Prob. 9P Ch. 5.3 - Prob. 10P Ch. 5.3 - Find a basis of solutions by the Frobenius method....Ch. 5.3 - Find a basis of solutions by the Frobenius method....Ch. 5.3 - Find a general solution in terms of hypergeometric...Ch. 5.3 - Find a general solution in terms of hypergeometric...Ch. 5.3 - Find a general solution in terms of hypergeometric...Ch. 5.3 - Find a general solution in terms of hypergeometric...Ch. 5.3 - Find a general solution in terms of hypergeometric...Ch. 5.3 - Find a general solution in terms of hypergeometric...Ch. 5.4 - Prob. 1P Ch. 5.4 - Prob. 2P Ch. 5.4 - Prob. 3P Ch. 5.4 - Prob. 4P Ch. 5.4 - Prob. 5P Ch. 5.4 - Prob. 6P Ch. 5.4 - Prob. 7P Ch. 5.4 - Prob. 8P Ch. 5.4 - Prob. 9P Ch. 5.4 - Prob. 10P Ch. 5.4 - Prob. 11P Ch. 5.4 - Prob. 12P Ch. 5.4 - Prob. 13P Ch. 5.4 - Prob. 14P Ch. 5.4 - Interlacing of zeros. Using (21) and Rolle’s...Ch. 5.4 - Prob. 16P Ch. 5.4 - Bessel’s equation. Show that for (1) the...Ch. 5.4 - Elementary Bessel functions. Derive (22) in...Ch. 5.4 - Prob. 19P Ch. 5.4 - Prob. 20P Ch. 5.4 - Use the powerful formulas (21) to do Probs. 19–25....Ch. 5.4 - Prob. 22P Ch. 5.4 - Use the powerful formulas (21) to do Probs. 19–25....Ch. 5.4 - Use the powerful formulas (21) to do Probs. 19–25....Ch. 5.4 - Use the powerful formulas (21) to do Probs. 19–25....Ch. 5.5 - Prob. 1P Ch. 5.5 - Prob. 2P Ch. 5.5 - Prob. 3P Ch. 5.5 - Prob. 4P Ch. 5.5 - Prob. 5P Ch. 5.5 - Prob. 6P Ch. 5.5 - Prob. 7P Ch. 5.5 - Prob. 8P Ch. 5.5 - Prob. 9P Ch. 5.5 - Hankel functions. Show that the Hankel functions...Ch. 5.5 - Modified Bessel functions of the first kind of...Ch. 5.5 - Prob. 13P Ch. 5.5 - Reality of Iv. Show that Iv(x) is real for all...Ch. 5.5 - Modified Bessel functions of the third kind...Ch. 5 - Prob. 1RQ Ch. 5 - What is the difference between the two methods in...Ch. 5 - Prob. 3RQ Ch. 5 - Prob. 4RQ Ch. 5 - Write down the most important ODEs in this chapter...Ch. 5 - Can a power series solution reduce to a...Ch. 5 - What is the hypergeometric equation? Where does...Ch. 5 - List some properties of the Legendre polynomials. Ch. 5 - Prob. 9RQ Ch. 5 - Can a Bessel function reduce to an elementary...Ch. 5 - POWER SERIES METHOD OR FROBENIUS METHOD Find a...Ch. 5 - POWER SERIES METHOD OR FROBENIUS METHOD Find a...Ch. 5 - POWER SERIES METHOD OR FROBENIUS METHOD Find a...Ch. 5 - Prob. 14RQ Ch. 5 - Prob. 15RQ Ch. 5 - Prob. 16RQ Ch. 5 - Prob. 17RQ Ch. 5 - Prob. 18RQ Ch. 5 - Prob. 19RQ Ch. 5 - Prob. 20RQ

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