Explanation Given information: Given differential equation is y ' = 1 − y 2 1 + x 2 On solving the given differential equation y ' = 1 − y 2 1 + x 2 y ' 1 − y 2 = 1 1 + x 2 ∫ d y 1 − y 2 = ∫ d x 1 + x 2 sin − 1 y = tan − 1 x + C y = sin ( tan − 1 x + C ) Now for expressing above solution in given form, use addition formula for sine function and solving sin ( tan − 1 x + C ) = sin ( tan − 1 x ) cos C + cos ( tan − 1 x ) sin C ......... ( 1 ) Note that sec 2 θ = 1 + tan 2 θ cos 2 θ = 1 1 + tan 2 θ cos θ = 1 1 + tan 2 θ .......... ( 2 ) Also 1 − sin 2 θ = 1 1 + tan 2 θ sin 2 θ = 1 − 1 1 + tan 2 θ sin 2 θ = tan 2 θ 1 + tan 2 θ sin θ = tan θ 1 + tan 2 θ

4th Edition

ISBN: 9781319308858

Author: Rogawski

Publisher: MAC HIGHER

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Differential Equations

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Chapter 9, Problem 49CRE

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To find the general solution of given differential equation and express the general solution in the given form.

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Chapter 9, Problem 48CRE

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Chapter 9 Solutions

CALCULUS:EARLY TRANS.(LL)-W/ACCESS

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To find the general solution of given differential equation and express the general solution in the given form.

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Chapter 9 Solutions