Explanation Given information: The given differential equation is y ' + k 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 Now, the given differential equation is, y ' + k y = 0 Inserting the value of y and y ' in above equation, ∑ n = 1 ∞ n a n x n − 1 + k ∑ n = 0 ∞ a n x n = 0 ∑ n = 1 ∞ n a n x n − 1 = − k ∑ n = 0 ∞ a n x n To obtain equal power of x , adjust the summation indices by replacing n by n +1 in the left hand side sum to obtain, ∑ n = 0 ∞ ( n + 1 ) a n + 1 x n = − k ∑ n = 0 ∞ a n x n Equating the coefficients of like terms, one can obtain the recursion formula

11th Edition

ISBN: 9781337275378

Author: Ron Larson, Bruce H. Edwards

Publisher: Cengage Learning

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Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differentia…

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Chapter 16.4, Problem 16E

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

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Chapter 16.4, Problem 15E

Chapter 16.4, Problem 17E

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Chapter 16 Solutions

Multivariable Calculus

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

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

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Chapter 16 Solutions