Explanation Given information: The given differential equation is y ″ + k 2 y = 0 . Formula used: Power series is defined as an infinite series of the form, ∑ n = 0 ∞ a n x n = a 0 + a 1 x + a 2 x 2 + a 3 x 3 + ......... + a n x n + .... , where x is a variable. Proof: Assume y = ∑ n = 0 ∞ a n x n is a solution of the differential equation. Then, differentiating the assumed solution with respect to x , y ' = ∑ n = 1 ∞ n a n x n − 1 y ″ = ∑ n = 2 ∞ n ( n − 1 ) a n x n − 2 Now, the given differential equation is, y ″ + k 2 y = 0 Inserting the value of y and y ″ in above equation, ∑ n = 2 ∞ n ( n − 1 ) a n x n − 2 + k 2 ∑ n = 0 ∞ a n x n = 0 ∑ n = 2 ∞ n ( n − 1 ) a n x n − 2 = − k 2 ∑ n = 0 ∞ a n x n To obtain equal power of x , adjust the summation indices by replacing n by n +2 in the left hand side sum to obtain, ∑ n = 0 ∞ ( n + 2 ) ( n + 1 ) a n + 2 x n = − k 2 ∑ n = 0 ∞ a n x n Equating the coefficients of like terms, one can obtain the recursion formula

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Author: Larson

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Chapter 16.4, Problem 18E

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To verify: That the solution of the differential equation y″+k2y=0 is same as obtained by using a power series and by any other method.

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Chapter 16.4, Problem 17E

Chapter 16.4, Problem 19E

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To verify: That the solution of the differential equation y ″ + k 2 y = 0 is same as obtained by using a power series and by any other method.

Solution Summary: The author explains how to verify that the solution of the differential equation is the same as obtained by using a power series and any other method.

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To verify: That the solution of the differential equation y″+k2y=0 is same as obtained by using a power series and by any other method.

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