Explanation Given information: The given differential equation is y ' ' − ( x − 1 ) y − y = 0 . Concept 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. Calculation: Assume y = ∑ n = 0 ∞ a n ( x − 1 ) n is a solution of the differential equation. Then, differentiating the assumed solution with respect to x , y ' = ∑ n = 1 ∞ n a n ( x − 1 ) n − 1 Again differentiating above equation with respect to x , y ' ' = ∑ n = 2 ∞ n ( n − 1 ) a n ( x − 1 ) n − 2 Now, the given differential equation is, y ' ' − ( x − 1 ) y − y = 0 Inserting the values of y and y ' ' in above equation, y ' ' − ( x − 1 ) y − y = 0 ∑ n = 2 ∞ n ( n − 1 ) a n ( x − 1 ) n − 2 − ( x − 1 ) ∑ n = 0 ∞ a n ( x − 1 ) n − ∑ n = 0 ∞ a n ( x − 1 ) n = 0 ∑ n = 2 ∞ n ( n − 1 ) a n ( x − 1 ) n − 2 − ∑ n = − 1 ∞ a n ( x − 1 ) n + 1 − ∑ n = 0 ∞ a n ( x − 1 ) n = 0 ∑ n = 2 ∞ n ( n − 1 ) a n ( x − 1 ) n − 2 = ∑ n = − 1 ∞ a n ( x − 1 ) n + 1 + ∑ n = 0 ∞ a n ( x − 1 ) n ∑ n = 0 ∞ ( n + 2 ) ( n + 1 ) a n + 2 ( x − 1 ) n = ∑ n = 0 ∞ a n − 1 ( x − 1 ) n + ∑ n = 0 ∞ a n ( x − 1 ) n Equating the coefficients of like terms, one can obtain the recursion formula. ( n + 2 ) ( n + 1 ) a n + 2 = a n − 1 + a n Which implies that, a n + 2 = a n − 1 ( n + 2 ) ( n + 1 ) + a n ( n + 2 ) ( n + 1 ) , n ≥ 0 Above formula will generate the following results, a 0 a 1 a 2 = a 0 − 1 ( 0 + 2 ) ( 0 + 1 ) + a 0 ( 0 + 2 ) ( 0 + 1 ) = a 0 2 ! a 3 = a 1 − 1 ( 1 + 2 ) ( 1 + 1 ) + a 1 ( 1 + 2 ) ( 1 + 1 ) = a 0 3 ! + a 1 3 ! = a 0 + a 1 3 ! a 4 = a 2 − 1 ( 2 + 2

Student Solutions Manual For Larson/edwards? Multivariable Calculus, 11th

11th Edition

ISBN: 9781337275392

Author: Larson, Ron; Edwards, Bruce H.

Publisher: Cengage Learning

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Differential Equations

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Chapter 16, Problem 15PS

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To calculate: The first eight terms of the solution of the Airy’sequation, y''−(x−1)y−y=0, using a power series.

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Chapter 16, Problem 14PS

Chapter 16, Problem 16PS

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Chapter 16 Solutions

Student Solutions Manual For Larson/edwards? Multivariable Calculus, 11th

Ch. 16.1 - Exactness What does it mean for the...Ch. 16.1 - Integrating Factor When is it beneficial to use an...Ch. 16.1 - Testing for Exactness In Exercises 3-6, determine...Ch. 16.1 - Testing for Exactness In Exercises 3-6, determine...Ch. 16.1 - Prob. 5E Ch. 16.1 - Prob. 6E Ch. 16.1 - Prob. 7E Ch. 16.1 - Solving an Exact Differential Equation In...Ch. 16.1 - Prob. 9E Ch. 16.1 - Prob. 10E

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The first eight terms of the solution of the Airy’sequation, y ' ' − ( x − 1 ) y − y = 0 , using a power series.

Solution Summary: The author calculates the first eight terms of the solution of Airy's equation using a power series.

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To calculate: The first eight terms of the solution of the Airy’sequation, y''−(x−1)y−y=0, using a power series.

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Chapter 16 Solutions