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6) Let y(x)=Σn-anx be a series solution of the nonhomogeneous ODE =0 y" - xy' - y = g(x), expanded about the ordinary point x0 = 0, where g(x) = Σn_09nx". =0

6) Let y(x)=Σn-anx be a series solution of the nonhomogeneous ODE =0 y" - xy' - y = g(x), expanded about the ordinary point x0 = 0, where g(x) = Σn_09nx". =0

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter2: Equations And Inequalities
Section2.1: Equations
Problem 62E
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please make a nice detailed solution it will help so much thank you

6) Let y(x) = 0 anx be a series solution of the nonhomogeneous ODE expanded about the ordinary point x0 = - y" — xy' – y = g(x), 0, where g(x) =Σn=0 Inx". -0 (a) Show that the coefficients an satisfy (n+2)(n+1)an+2 − (n + 1)an = In n≥ 0. (b) When g(x) = 1 + 2x, find the coefficients a2, a3, a4, a5 and a6 and hence write down the power series of the solution y(x) up to the term involving 26, where the only arbitrary constants should be ao and a₁.
Transcribed Image Text:6) Let y(x) = 0 anx be a series solution of the nonhomogeneous ODE expanded about the ordinary point x0 = - y" — xy' – y = g(x), 0, where g(x) =Σn=0 Inx". -0 (a) Show that the coefficients an satisfy (n+2)(n+1)an+2 − (n + 1)an = In n≥ 0. (b) When g(x) = 1 + 2x, find the coefficients a2, a3, a4, a5 and a6 and hence write down the power series of the solution y(x) up to the term involving 26, where the only arbitrary constants should be ao and a₁.
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