Explanation Use the given equation to substitute n = 1 and then solve the integration by parts to obtain the proof Given: Let J n = ∫ x n e − x 2 / 2 d x Calculation: Let u = − x 2 2 . So d u = − x d x Setting n = 1 in the given equation, J n = ∫ x n e... To determine (b) To prove that J n = − x n − 1 e − x 2 / 2 + ( n − 1 ) J n − 2 To determine (c) To compute J 3 and J 5 using (a) and (b)

4th Edition

ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Differential Equations

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Question

Chapter 8, Problem 68CRE

To determine

(a)

To show that $J_{1} = - e^{- x^{2} / 2}$

To determine

(b)

To prove that $J_{n} = - x^{n - 1} e^{- x^{2} / 2} + (n - 1) J_{n - 2}$

To determine

(c)

To compute $J_{3}$ and $J_{5}$ using (a) and (b)

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Chapter 8, Problem 67CRE

Chapter 8, Problem 69CRE

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Chapter 8 Solutions

CALCULUS (CLOTH)

Ch. 8.1 - Prob. 1PQ Ch. 8.1 - Prob. 2PQ Ch. 8.1 - Prob. 3PQ Ch. 8.1 - Prob. 1E Ch. 8.1 - Prob. 2E Ch. 8.1 - Prob. 3E Ch. 8.1 - Prob. 4E Ch. 8.1 - Prob. 5E Ch. 8.1 - Prob. 6E Ch. 8.1 - Prob. 7E

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(a) To show that J 1 = − e − x 2 / 2

Solution Summary: The author explains that they have proved that J 1 = e x 2 / 2 using the given equation to substitute n = 1 and then solve the integration by parts to obtain the value of J

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(a)

To show that $J_{1} = - e^{- x^{2} / 2}$

(b)

To prove that $J_{n} = - x^{n - 1} e^{- x^{2} / 2} + (n - 1) J_{n - 2}$

(c)

To compute $J_{3}$ and $J_{5}$ using (a) and (b)

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Chapter 8 Solutions

(a) To show that J 1 = − e − x 2 / 2

Solution Summary: The author explains that they have proved that J 1 = e x 2 / 2 using the given equation to substitute n = 1 and then solve the integration by parts to obtain the value of J

Concept explainers

Videos

(a) To show that J1=−e−x2/2

(b) To prove that Jn=−xn−1e−x2/2+(n−1)Jn−2

(c) To compute J3 and J5 using (a) and (b)

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Chapter 8 Solutions

(a)

To show that $J_{1} = - e^{- x^{2} / 2}$

(b)

To prove that $J_{n} = - x^{n - 1} e^{- x^{2} / 2} + (n - 1) J_{n - 2}$

(c)

To compute $J_{3}$ and $J_{5}$ using (a) and (b)