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nd we are assuming k = 1.) " ia;2¹-1 1 ia,z¹-1 5 iaz¹-1= 1 i=1 1 1-0 i-0 9;x¹+1 9;x¹ dia1 - Σα (i+1)ia,z¹+¹ = ∞ i=1 aiz¹+1

nd we are assuming k = 1.) " ia;2¹-1 1 ia,z¹-1 5 iaz¹-1= 1 i=1 1 1-0 i-0 9;x¹+1 9;x¹ dia1 - Σα (i+1)ia,z¹+¹ = ∞ i=1 aiz¹+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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Substituting f'(x) into the differential equation y' = ky, which of the following must be satisfied in order for f(x) = Σºa;x² to be a solution of the differential equation? (Keep in mind we are assuming k = 1.) 8WI 8 i=0 O IM: ∞ Σ i=1 ia; x ia;x iax²-1 `α₁x² Σ+ i=1 IM8 8WI i=0 8 8 WI a;x² Aix²+1 `(i+1)ia¿x³−¹ aix² Ma i=1 aixi+1
Transcribed Image Text:Substituting f'(x) into the differential equation y' = ky, which of the following must be satisfied in order for f(x) = Σºa;x² to be a solution of the differential equation? (Keep in mind we are assuming k = 1.) 8WI 8 i=0 O IM: ∞ Σ i=1 ia; x ia;x iax²-1 `α₁x² Σ+ i=1 IM8 8WI i=0 8 8 WI a;x² Aix²+1 `(i+1)ia¿x³−¹ aix² Ma i=1 aixi+1
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.
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