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SOLVE ALL PARTS PLEASE

SOLVE ALL PARTS PLEASE

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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G SOLVE ALL PARTS PLEASE

Previously, we learned some of the many important applications of differential equations, and learned some approaches to solve or analyze them. Here, we consider an important approach that will allow us to solve a wider variety of differential equations. Let's consider the familiar differential equation from exponential population growth given by y = ky, where k is the constant of proportionality. While we can solve this differential equation using familiar methods, we take a different approach now that can be applied to a much larger set of differential equations. For the rest of this activity, let's assume that k = 1. We will use our knowledge of Taylor series to find a solution to the differential equation. To do so, we assume that we have a solution y = f(x) and that f(x) has a Taylor series that can be written in the form ∞ y=f(x) = Σa;x¹, i=0 where the coefficients a; are undetermined. Our task is to find the coefficients.
Transcribed Image Text:Previously, we learned some of the many important applications of differential equations, and learned some approaches to solve or analyze them. Here, we consider an important approach that will allow us to solve a wider variety of differential equations. Let's consider the familiar differential equation from exponential population growth given by y = ky, where k is the constant of proportionality. While we can solve this differential equation using familiar methods, we take a different approach now that can be applied to a much larger set of differential equations. For the rest of this activity, let's assume that k = 1. We will use our knowledge of Taylor series to find a solution to the differential equation. To do so, we assume that we have a solution y = f(x) and that f(x) has a Taylor series that can be written in the form ∞ y=f(x) = Σa;x¹, i=0 where the coefficients a; are undetermined. Our task is to find the coefficients.
Part (a) Assume that we can differentiate a power series term by term. By taking the derivative of f(x) with respect to a. Which of the following is the derivative of f(x)? f'(x) = f'(x) υ f'(x) O Ο - f'(x) ∞ - Σ(1-1)a;ai i=0 f'(x) = Σiazi-1 - Σαρα 2=0 00 Σία π i=0 ∞ i=0 ∞ Σιαπι i=0
Transcribed Image Text:Part (a) Assume that we can differentiate a power series term by term. By taking the derivative of f(x) with respect to a. Which of the following is the derivative of f(x)? f'(x) = f'(x) υ f'(x) O Ο - f'(x) ∞ - Σ(1-1)a;ai i=0 f'(x) = Σiazi-1 - Σαρα 2=0 00 Σία π i=0 ∞ i=0 ∞ Σιαπι i=0
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