(a) To determine The solution for the initial value problem with first five nonzero terms. Explanation Given: Differential equation is y ″ − x y ′ − y = 0 with initial value y ( 0 ) = 2 and y ′ ( 0 ) = 1 . Calculation: From Problem 2, the solution of the differentiable equation y ″ − x y ′ − y = 0 at x 0 = 0 is y = a 0 [ 1 + x 2 2 + x 4 2 ⋅ 4 + x 6 2 ⋅ 4 ⋅ 6 + … ] + a 1 [ x + x 3 3 + x 5 3 ⋅ 5 + x 7 3 ⋅ 5 ⋅ 7 + … ] . Given initial conditions are y ( 0 ) = 2 and y ′ ( 0 ) = 1 Equate y ( 0 ) = 2 as follows. y ( 0 ) = 2 a 0 [ 1 + ( 0 ) 2 2 + ( 0 ) 4 2 ⋅ 4 + ( 0 ) 6 2 ⋅ 4 ⋅ 6 + … ] + a 1 [ 0 + ( 0 ) 3 3 + ( 0 ) 5 3 ⋅ 5 + ( 0 ) 7 3 ⋅ 5 ⋅ 7 + … ] = 2 a 0 ( 1 ) + a 1 ( 0 ) = 2 a 0 = 2 Thus, the value of a 0 is 2. Compute the derivative of y as follows. y ′ ( x ) = ( a 0 [ 1 + x 2 2 + x 4 2 ⋅ 4 + x 6 2 ⋅ 4 ⋅ 6 + … ] + a 1 [ x + x 3 3 + x 5 3 ⋅ 5 + x 7 3 ⋅ 5 ⋅ 7 + … ] ) ′ = a 0 ( 0 + 2 x 2 + 4 x 3 2 ⋅ 4 + 6 x 5 2 ⋅ 4 ⋅ 6 + … ) + a 1 ( 1 + 3 x 2 3 + 5 x 4 3 ⋅ 5 + 7 x 6 3 ⋅ 5 ⋅ 7 + … ) = a 0 ( x + x 3 2 + x 5 2 ⋅ 4 + … ) + a 1 ( 1 + x 2 + x 4 3 + x 6 3 ⋅ 5 + … ) Equate y ′ ( 0 ) = 1 as follows (b) To determine To plot: The four-term and the five-term approximation of the solution. (c) To determine The four-term approximation interval that is reasonably accurate.

10th Edition

ISBN: 9780470458327

Author: William E. Boyce, Richard C. DiPrima

Publisher: Wiley, John & Sons, Incorporated

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Chapter 5.2, Problem 15P

(a)

To determine

The solution for the initial value problem with first five nonzero terms.

(b)

To determine

To plot: The four-term and the five-term approximation of the solution.

(c)

To determine

The four-term approximation interval that is reasonably accurate.

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Chapter 5.2, Problem 14P

Chapter 5.2, Problem 16P

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Elementary Differential Equations

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