In the lectures, we constructed a series solution centered at the origin for the Airy's equation...

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In the lectures, we constructed a series solution centered at the origin for the Airy's equation y" - xy = 0. Note that the coefficients in the Airy's equation are real-analytic everywhere on R. The objective of this question is to build power series solution centered at 1. More precisely, we look for a solution of the form y(x) = [an(x-1)" n=0 (2a) Write a deduce a series representation for y"(x) whose generic term is (x-1)". (2b) Rewriting x as 1+ (x-1), deduce a series representation for xy(x) whose generic term is (x-1)". (2c) Substituting your expressions from (2a) and (2b) in the Airy's equation, arrive at a recurrence relation for the coefficients an and hence determine these coefficients. (2d) Using the coefficients a determined in (2c), write down the expression for your series solution centered at 1. Further, show that it has the structure of a linear combination of two linearly independent solutions. (2e) Using ratio test, show that the series representations of your two linearly independent solutions converge for all x € R. and (2f) Recall that the series solutions (centered at the origin) for the Airy's equation that we obtained in lectures were 1 00 x³n y₁(x)=1+2.3... (3n-1)-(3n) n=1 DO y2(x) = x + Σ n=1 x³n+1 3.4 (3n) (3n+1)* Show, using ratio test, that both these series converge for all x € R. . In the lectures, we constructed a series solution centered at the origin for the Airy's equation y" - xy = 0. Note that the coefficients in the Airy's equation are real-analytic everywhere on R. The objective of this question is to build power series solution centered at 1. More precisely, we look for a solution of the form y(x) = [an(x-1)" n=0 (2a) Write a deduce a series representation for y"(x) whose generic term is (x-1)". (2b) Rewriting x as 1+ (x-1), deduce a series representation for xy(x) whose generic term is (x-1)". (2c) Substituting your expressions from (2a) and (2b) in the Airy's equation, arrive at a recurrence relation for the coefficients an and hence determine these coefficients. (2d) Using the coefficients a determined in (2c), write down the expression for your series solution centered at 1. Further, show that it has the structure of a linear combination of two linearly independent solutions. (2e) Using ratio test, show that the series representations of your two linearly independent solutions converge for all x € R. and (2f) Recall that the series solutions (centered at the origin) for the Airy's equation that we obtained in lectures were 1 00 x³n y₁(x)=1+2.3... (3n-1)-(3n) n=1 DO y2(x) = x + Σ n=1 x³n+1 3.4 (3n) (3n+1)* Show, using ratio test, that both these series converge for all x € R. .

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