Differential Equations for Engineers · Linear higher-order equations and oscillatory response
Consider the initial-value problem (x^2 + 9) y'' + 5x y' − 4y = 0, y(0) = 2, y'(0) = −1
Problem
Consider the initial-value problem (x^2 + 9) y'' + 5x y' − 4y = 0, y(0) = 2, y'(0) = −1. The point x = 0 is an ordinary point. Seek a power-series solution y(x) = Σ_{n=0}^∞ a_n x^n about x = 0. Determine the recurrence relation for the coefficients a_n (n ≥ 2), compute a_0 through a_6 explicitly, and write the sixth-degree Taylor polynomial for y about x = 0.
Hint
Insert \(y=\sum a_n x^n\) into the DE and shift every sum to the common power \(x^n\).
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