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

6th Edition

ISBN: 9781284105902

Author: Dennis G. Zill

Publisher: Jones & Bartlett Learning

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Differential Equations

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Chapter 3.7, Problem 23E

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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 3.7, Problem 22E

Chapter 3.8, Problem 1E

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For each month of the year, Taylor collected the average high temperatures in Jackson, Mississippi. He used the data to create the histogram shown. Which set of data did he use to create the histogram? A 55, 60, 64, 72, 73, 75, 77, 81, 83, 91, 91, 92\ 55,\ 60,\ 64,\ 72,\ 73,\ 75,\ 77,\ 81,\ 83,\ 91,\ 91,\ 92 55, 60, 64, 72, 73, 75, 77, 81, 83, 91, 91, 92 B 55, 57, 60, 65, 70, 71, 78, 79, 85, 86, 88, 91\ 55,\ 57,\ 60,\ 65,\ 70,\ 71,\ 78,\ 79,\ 85,\ 86,\ 88,\ 91 55, 57, 60, 65, 70, 71, 78, 79, 85, 86, 88, 91 C 55, 60, 63, 64, 65, 71, 83, 87, 88, 88, 89, 93\ 55,\ 60,\ 63,\ 64,\ 65,\ 71,\ 83,\ 87,\ 88,\ 88,\ 89,\ 93 55, 60, 63, 64, 65, 71, 83, 87, 88, 88, 89, 93 D 55, 58, 60, 66, 68, 75, 77, 82, 86, 89, 91, 91\ 55,\ 58,\ 60,\ 66,\ 68,\ 75,\ 77,\ 82,\ 86,\ 89,\ 91,\ 91 55, 58, 60, 66, 68, 75, 77, 82, 86, 89, 91, 91

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Chapter 3 Solutions

Advanced Engineering Mathematics

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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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