Explanation Given: The given differential equation is d y d x = − y a 2 − y 2 . Initial point that satisfies the given differential equation is ( 0 , 10 ) and the length of the rope is 10 ft . Calculation: Substitute 10 ft for a in the given differential equation. d y d x = − y 10 2 − y 2 d y d x = − y 100 − y 2 (1) Suppose that, y = 10 sin θ (2) Differentiate equation (2). d y = d ( 10 sin θ ) d y = 10 cos θ d θ General relation of trigonometry is, 10 2 sin 2 θ + 10 2 cos 2 θ = 100 100 − 10 2 sin 2 θ = 10 2 cos 2 θ 10 cos θ = 100 − 10 2 sin 2 θ (3) Substitute y for 10 sin θ in equation (3). 10 cos θ = 100 − y 2 Substitute 10 sin θ for y , 10 cos θ d θ for d y and 10 cos θ for 100 − y 2 in equation (1). 10 cos θ d θ d x = − 10 sin θ 10 cos θ 10 cos θ d θ d x = − sin θ cos θ (4) Separate the variables of the differential equation in equation (4), − 10 cos 2 θ sin θ d θ = d x − 10 ( 1 − sin 2 θ sin θ ) d θ = d x − 10 ( csc θ − sin θ ) d θ = d x (5) Integrate equation (5), − ∫ 10 ( csc θ − sin θ ) d θ = ∫ d x − 10 [ − ln | csc θ + cot θ | + cos θ ] = x + C (6) Find the values of the terms that are in θ in the form of x and y

6th Edition

ISBN: 9781284105902

Author: Dennis G. Zill

Publisher: Jones & Bartlett Learning

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Differential Equations

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Chapter 2 Solutions

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