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# Find two power series solutions of the given differential equation about the ordinary point x =...

Find two power series solutions of the given differential equation about the ordinary point x = 0. y′′ − 4xy′ + y = 0 Find two power series solutions of the given differential equation about the ordinary point x = 0. y!' - 4xy' + y = 0 Step 1 We are asked to find two power series solutions to the following homogenous linear second-order differential equation. y" - 4xy' + y = 0 By Theorem 6.2.1, we know two such solutions exist about the ordinary point x = 0. In fact, the differential equation has no singular points as each of the coefficient functions are very simple) polynomials. Therefore, they are trivially represented as power series at any point and are therefore analytic. This means the power series solution will converge for 1x1 < 0. Let y = n = 0 solve for two sets of cn to find the two solutions. to cox". We substitute this series and its derivatives into the given differential equation. We will The differential equation includes non-zero terms for the first and second derivative of y. These derivatives always have the same form, so we refer to equation (1) of section 6.1. y' = ☺ canxa - 1 n = 1 y" = n(n - 1)x7 - 2 n = 2 Substitute the power series into the given differential equation. y" - 4xy' + y = Š (c, n(n − 1)x9 2) - 4 (C (CMX") n = 2 n=1 n = 0 Submit Skip (you cannot come back)

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