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Problem #7: Consider the differential equation 2x²y" + (x² + x)y' - y = 0. (a) Without solving this differential equation, one can say that any power series solution centered at the ordinary point xo = 3 (i.e. a series involving powers of x − 3) has a radius of convergence at least equal to R. What is R? (Enter infinity if R = ∞.) (b) Find the two roots of the indicial equation associated with the regular singular point xo = 0. Separate your answers with a comma. (c) If r₁ denotes the largest of the 2 roots of the indicial equation, the differential equation above has a solution of the form ∞ Σ cnxntr n=0 where Cn = g(n)cn-1, n ≥ 1, for a certain function g(n). Compute g(n). (d) Write the sum of the first 3 non-zero terms of the series in (c) if co = 1.

Problem #7: Consider the differential equation 2x²y" + (x² + x)y' - y = 0. (a) Without solving this differential equation, one can say that any power series solution centered at the ordinary point xo = 3 (i.e. a series involving powers of x − 3) has a radius of convergence at least equal to R. What is R? (Enter infinity if R = ∞.) (b) Find the two roots of the indicial equation associated with the regular singular point xo = 0. Separate your answers with a comma. (c) If r₁ denotes the largest of the 2 roots of the indicial equation, the differential equation above has a solution of the form ∞ Σ cnxntr n=0 where Cn = g(n)cn-1, n ≥ 1, for a certain function g(n). Compute g(n). (d) Write the sum of the first 3 non-zero terms of the series in (c) if co = 1.

Advanced Engineering Mathematics
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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Problem #7: Consider the differential equation 2x²y" + (x² + x)y′ − y = 0. (a) Without solving this differential equation, one can say that any power series solution centered at the ordinary 3 (i.e. a series involving powers of x − 3) has a radius of convergence at least equal to R. What is R? (Enter infinity if R = ∞.) point xo = (b) Find the two roots of the indicial equation associated with the regular singular point xo answers with a comma. n=0 (c) If r₁ denotes the largest of the 2 roots of the indicial equation, the differential equation above has a solution of the form where Cn = Cnxn+r1 = 0. Separate your g(n)cn-1, n ≥ 1, for a certain function g(n). Compute g(n). (d) Write the sum of the first 3 non-zero terms of the series in (c) if co = 1. (e) What is the radius of convergence of the series in (c) if co = 1? (Enter infinity if it is infinite.)
Transcribed Image Text:Problem #7: Consider the differential equation 2x²y" + (x² + x)y′ − y = 0. (a) Without solving this differential equation, one can say that any power series solution centered at the ordinary 3 (i.e. a series involving powers of x − 3) has a radius of convergence at least equal to R. What is R? (Enter infinity if R = ∞.) point xo = (b) Find the two roots of the indicial equation associated with the regular singular point xo answers with a comma. n=0 (c) If r₁ denotes the largest of the 2 roots of the indicial equation, the differential equation above has a solution of the form where Cn = Cnxn+r1 = 0. Separate your g(n)cn-1, n ≥ 1, for a certain function g(n). Compute g(n). (d) Write the sum of the first 3 non-zero terms of the series in (c) if co = 1. (e) What is the radius of convergence of the series in (c) if co = 1? (Enter infinity if it is infinite.)
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