Explanation The Radius of convergence for any series basically tells about the domain on which that series is convergent. The Radio Test for series is very useful test for determining whether or not a series converges or diverges. The Ratio Test says that: Let the following limit exists, L = lim n → ∞ | a n + 1 a n | i. If L < 1 , then the original series ∑ n = 1 ∞ a n will be absolutely convergent. ii. If L > 1 or L = ∞ , then the original series ∑ n = 1 ∞ a n will be divergent. iii. If L = 1 , then no conclusion can be made (the series may diverge or converge). Given: The differential equation y ' ' − x y ' + y = 0 with initial condition is y ( 0 ) = 1 , y ' ( 0 ) = 0 . Formula Used: L = lim n → ∞ | a n + 1 a n | Calculation: Let y = P ( x ) = ∑ n = 0 ∞ a n x n y = a 0 + a 1 x + a 2 x 2 + ... .... ( 1 ) Now differentiate the equation (1) with respect to x and we get: y ' = P ' ( x ) = ∑ n = 1 ∞ n a n x n − 1 = a 1 + 2 a 2 x + 3 a 3 x 2 + ... And y ' ' = P ' ' ( x ) = ∑ n = 2 ∞ n ( n − 1 ) a n x n − 2 = 2 a 2 x + 6 a 2 x + ... Here, we can see that, y ( 0 ) = P ( 0 ) = a 0 , y ' ( 0 ) = P ' ( 0 ) = a 1 In order to satisfy the initial condition y ( 0 ) = 1 , y ' ( 0 ) = 0 , we must have a 0 = 1 , a 1 = 0 . Now, the differential equation is: y ' ' − x y ' + y = 0 ... ( 2 ) Now, substitute the values of y,y’and y’’into the equation (2) and we get: y ' ' − x y ' + y = ∑ n = 2 ∞ n ( n − 1 ) a n x n − 2 − x ∑ n = 1 ∞ n a n x n − 1 + ∑ n = 0 ∞ a n x n = ∑ n = 2 ∞ n ( n − 1 ) a n x n − 2 − ∑ n = 1 ∞ n a n x n + ∑ n = 0 ∞ a n x n = ∑ n + 2 = 2 ∞ ( n

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Chapter 11.6, Problem 65E

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A power series $P (x)$ satisfying the differential equation $y'' - x y' + y = 0$ with initial condition $y (0) = 1, y' (0) = 0$ . Then find the radius of convergence of the power series.

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Chapter 11.6, Problem 64E

Chapter 11.6, Problem 66E

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CALCULUS 4E (LL) W/ ACHIEVE ACCESS

Ch. 11.1 - Prob. 1PQ Ch. 11.1 - Prob. 2PQ Ch. 11.1 - Prob. 3PQ Ch. 11.1 - Prob. 4PQ Ch. 11.1 - Prob. 5PQ Ch. 11.1 - Prob. 1E Ch. 11.1 - Prob. 2E Ch. 11.1 - Prob. 3E Ch. 11.1 - Prob. 4E Ch. 11.1 - Prob. 5E

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A power series P ( x ) satisfying the differential equation y ' ' − x y ' + y = 0 with initial condition y ( 0 ) = 1 , y ' ( 0 ) = 0 . Then find the radius of convergence of the power series.

Solution Summary: The author explains that the Radius of convergence for any series tells about the domain on which that series is convergent.

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To calculate:

A power series $P (x)$ satisfying the differential equation $y'' - x y' + y = 0$ with initial condition $y (0) = 1, y' (0) = 0$ . Then find the radius of convergence of the power series.

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A power series P ( x ) satisfying the differential equation y ' ' − x y ' + y = 0 with initial condition y ( 0 ) = 1 , y ' ( 0 ) = 0 . Then find the radius of convergence of the power series.

Solution Summary: The author explains that the Radius of convergence for any series tells about the domain on which that series is convergent.

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To calculate: A power series P(x) satisfying the differential equation y''−xy'+y=0 with initial condition y(0)=1,y'(0)=0. Then find the radius of convergence of the power series.

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Chapter 11 Solutions

To calculate:

A power series $P (x)$ satisfying the differential equation $y'' - x y' + y = 0$ with initial condition $y (0) = 1, y' (0) = 0$ . Then find the radius of convergence of the power series.