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In this exercise we consider finding the first five coefficients in the series solution of the first order linear initial value problem (x + 2) y' + 3y = 0 subject to the initial condition y(0) = 1. Since the equation has an ordinary point at x = 0 it has a power series solution in the form We learned how to easily solve problems like this separation of variables but here we want to consider the power series method. (1) Insert the formal power series into the differential equation and derive the recurrence relation an = -n/(n+1) (2) Use the recurrence relation to find the first few coefficients , a₁ = -1/2 ao 1 1 an = (3) Write out a few more terms if needed and try to guess the pattern for the general term for all n = 1, 2,... (4) Therefore the solution can be written as ,a₂ = -1/2 Notice The general solution of this first order linear equation is y IL , az = 4 (x + 2)² y = y = anx 2 n=0 2 n=0 an-1 for n = 1, 2,... 2.²

In this exercise we consider finding the first five coefficients in the series solution of the first order linear initial value problem (x + 2) y' + 3y = 0 subject to the initial condition y(0) = 1. Since the equation has an ordinary point at x = 0 it has a power series solution in the form We learned how to easily solve problems like this separation of variables but here we want to consider the power series method. (1) Insert the formal power series into the differential equation and derive the recurrence relation an = -n/(n+1) (2) Use the recurrence relation to find the first few coefficients , a₁ = -1/2 ao 1 1 an = (3) Write out a few more terms if needed and try to guess the pattern for the general term for all n = 1, 2,... (4) Therefore the solution can be written as ,a₂ = -1/2 Notice The general solution of this first order linear equation is y IL , az = 4 (x + 2)² y = y = anx 2 n=0 2 n=0 an-1 for n = 1, 2,... 2.²

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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In this exercise we consider finding the first five coefficients in the series solution of the first order linear initial value problem (x + 2) y' + 3y = 0 subject to the initial condition y(0) = 1. Since the equation has an ordinary point at = 0 it has a power series solution in the form (2) Use the recurrence relation to find the first few coefficients ao = We learned how to easily solve problems like this separation of variables but here we want to consider the power series method. (1) Insert the formal power series into the differential equation and derive the recurrence relation an 1 = a 1 = -1/2 a2 (4) Therefore the solution can be written as -1/2 (3) Write out a few more terms if needed and try to guess the pattern for the general term for all n = 1, 2, .. az Notice The general solution of this first order linear equation is y an 4 (x + 2)² y = -n/(n+1) || ∞ ] y = n=0 ∞ n=0 Anxn an-1 for n = 1, 2,... xn
Transcribed Image Text:In this exercise we consider finding the first five coefficients in the series solution of the first order linear initial value problem (x + 2) y' + 3y = 0 subject to the initial condition y(0) = 1. Since the equation has an ordinary point at = 0 it has a power series solution in the form (2) Use the recurrence relation to find the first few coefficients ao = We learned how to easily solve problems like this separation of variables but here we want to consider the power series method. (1) Insert the formal power series into the differential equation and derive the recurrence relation an 1 = a 1 = -1/2 a2 (4) Therefore the solution can be written as -1/2 (3) Write out a few more terms if needed and try to guess the pattern for the general term for all n = 1, 2, .. az Notice The general solution of this first order linear equation is y an 4 (x + 2)² y = -n/(n+1) || ∞ ] y = n=0 ∞ n=0 Anxn an-1 for n = 1, 2,... xn
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