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t (e) ite a4 in terms of a3. Then write a4 in terms of ao. When writing your answer, use a3 for a3 wer in terms of a3: a4= =wer in terms of ao: a4 = t (f) serve that there is a pattern in (b)-(e). Find a general formula for a in terms of ao. Use this formula to calculate a101 in terms of a. =wer in terms of ag: a101 1. ng your formula for a; in part (f), write the series expansion for y using only the unknown coefficient ao. Factor out ag and determine what familiar functions satisfy the differential ation y' = ky. (Hint: Compare to a familiar Taylor series.) wer: y

t (e) ite a4 in terms of a3. Then write a4 in terms of ao. When writing your answer, use a3 for a3 wer in terms of a3: a4= =wer in terms of ao: a4 = t (f) serve that there is a pattern in (b)-(e). Find a general formula for a in terms of ao. Use this formula to calculate a101 in terms of a. =wer in terms of ag: a101 1. ng your formula for a; in part (f), write the series expansion for y using only the unknown coefficient ao. Factor out ag and determine what familiar functions satisfy the differential ation y' = ky. (Hint: Compare to a familiar Taylor series.) wer: y

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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G3solve 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 (e) Write a4 in terms of a3. Then write a in terms of ao. When writing your answer, use a3 for a3 Answer in terms of a3: a4 Answer in terms of ao: a4 Part (f) Observe that there is a pattern in (b)-(e). Find a general formula for a in terms of ao. Use this formula to calculate a 101 in terms of ao. Answer in terms of ag: a101 Part (g) Using your formula for a¿ in part (f), write the series expansion for y using only the unknown coefficient ao. Factor out ao and determine what familiar functions satisfy the differential equation y' = ky. (Hint: Compare to a familiar Taylor series.) Answer: y
Transcribed Image Text:Part (e) Write a4 in terms of a3. Then write a in terms of ao. When writing your answer, use a3 for a3 Answer in terms of a3: a4 Answer in terms of ao: a4 Part (f) Observe that there is a pattern in (b)-(e). Find a general formula for a in terms of ao. Use this formula to calculate a 101 in terms of ao. Answer in terms of ag: a101 Part (g) Using your formula for a¿ in part (f), write the series expansion for y using only the unknown coefficient ao. Factor out ao and determine what familiar functions satisfy the differential equation y' = ky. (Hint: Compare to a familiar Taylor series.) Answer: y
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