Explanation Formula used: A first-order differential equation of the form d y d x = g ( x ) h ( y ) is said to be separable or to have separable variables. Calculation: A mathematical model for the position x ( t ) of a body moving rectilinearly on the x -axis in an inverse-square force field is given by d 2 x d t 2 = − k 2 x 2 . Let u = d x d t , d 2 x d t 2 = u ⋅ d u d x . So, given equation becomes, u d u d x = − k 2 x 2 Separate the variables u ⋅ d u = − k 2 x 2 d x Integrate both sides 1 2 u 2 = k 2 x + c 1 2 v 2 = k 2 x + c When t = 0 the body starts from rest from the position x = x 0 , and v = 0 . So, 0 = k 2 x 0 + c c = − k 2 x 0 1 2 v 2 = k 2 x − k 2 x 0 v 2 = 2 k 2 ( 1 x − 1 x 0 ) Hence proved u = − 2 k x 0 − x

Differential Equations with Boundary-Value Problems (MindTap Course List)

9th Edition

ISBN: 9781305965799

Author: Dennis G. Zill

Publisher: Cengage Learning

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Differential Equations

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Chapter 4.10, Problem 25E

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To show: The velocity of the body at time t is given by v2=2k2(1/x−1/x0).

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Chapter 4.10, Problem 24E

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Differential Equations with Boundary-Value Problems (MindTap Course List)

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The velocity of the body at time t is given by v 2 = 2 k 2 ( 1 / x − 1 / x 0 ) .

Solution Summary: The author explains the mathematical model for the position x of a body moving rectilinearly on the X -axis in an inverse-square force field.

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To show: The velocity of the body at time t is given by v2=2k2(1/x−1/x0).

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Chapter 4 Solutions