a. Explanation Given: The differential equation y ′ ( t ) + 4 y = 8 , y ( 0 ) = 0 . Calculation: Assume that the solution has the a Taylor series centered at 0 of the form y ( t ) = ∑ k = 0 ∞ c k t k and the coefficient of the Taylor series c k = y ( k ) ( 0 ) k ! , for k = 0 , 1 , 2 , ... . Substitute 0 for t in y ′ ( t ) + 4 y = 8 and use the initial condition y ( 0 ) = 0 , y ′ ( 0 ) + 4 y ( 0 ) = 8 y ′ ( 0 ) = 8 − 4 y ( 0 ) y ′ ( 0 ) = 8 − 4 ⋅ 0 ( ∵ y ( 0 ) = 0 ) y ′ ( 0 ) = 8 Thar is, y ′ ( 0 ) = 8 . Differentiate y ′ ( t ) + 4 y = 8 as follows, y ″ ( t ) + 4 y ′ ( t ) = 0 y ″ ( t ) = − 4 y ′ ( t ) y ″ ( 0 ) = − 4 ( 8 ) ( ∵ y ′ ( 0 ) = 8 ) y ″ ( 0 ) = 8 ( − 4 ) Again differentiate y ″ ( t ) + 4 y ′ ( t ) = 0 as follows, y ‴ ( t ) + 4 y ″ ( t ) = 0 y ‴ ( t ) = − 4 y ″ ( t ) y ‴ ( t ) = − 4 ( 8 ( − 4 ) ) ( ∵ y ″ ( 0 ) = 8 ( − 4 ) ) y ‴ ( t ) = 8 ( − 4 ) 2 In general, y k ( 0 ) = 8 ( − 4 ) k − 1 , k = 1 , 2 , b. To determine To identify: The function represented by the power series.

Calculus, Single Variable: Early Transcendentals (3rd Edition)

3rd Edition

ISBN: 9780134766850

Author: William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz

Publisher: PEARSON

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Chapter 11.4, Problem 34E

To determine

To find: The power series for the solution of the given differential equation, subject to the initial condition.

To determine

To identify: The function represented by the power series.

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Chapter 11.4, Problem 33E

Chapter 11.4, Problem 35E

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Chapter 11 Solutions

Calculus, Single Variable: Early Transcendentals (3rd Edition)

Ch. 11.1 - Verify that p3 satisfies p3(k)(a)=f(k)(a), for k =...Ch. 11.1 - Prob. 2QC Ch. 11.1 - Prob. 3QC Ch. 11.1 - Write out the next two Taylor polynomials p4 and...Ch. 11.1 - Prob. 5QC Ch. 11.1 - Prob. 6QC Ch. 11.1 - Suppose you use a second-order Taylor polynomial...Ch. 11.1 - Does the accuracy of an approximation given by a...Ch. 11.1 - The first three Taylor polynomials for f(x)=1+x...Ch. 11.1 - Prob. 4E

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The power series for the solution of the given differential equation, subject to the initial condition.

Solution Summary: The author calculates the power series for the solution of the given differential equation, subject to the initial condition.