Explanation Given: The first order differential equation is given by, d X d t = ( X − 1 ) ( 1 − 2 X ) Definition used: “A relation G ( x , y ) = 0 is said to be an implicit solution of an ordinary differential equation on an interval I , provided that there exists at least one function ϕ that satisfies the relation as well as the differential equation on I .” Calculation: Consider the function ln ( 2 X − 1 X − 1 ) = t . The function can be written as, ln ( 2 X − 1 ) − ln ( X − 1 ) = t Differentiate the given function with respect to t . d d t ( ln ( 2 X − 1 ) ) − d d t ( ln ( X − 1 ) ) = d d t ( t ) 1 ( 2 X − 1 ) d d t ( 2 X − 1 ) − 1 ( X − 1 ) d d t ( X − 1 ) = 1 1 ( 2 X − 1 ) 2 ( 1 ) d X d t − 1 ( X − 1 ) d X d t = 1 d X d t ( 2 ( 2 X − 1 ) − 1 ( X − 1 ) ) = 1 Simplify further, d X d t ( 2 X − 2 − 2 X + 1 ( 2 X − 1 ) ( X − 1 ) ) = 1

6th Edition

ISBN: 9781284105902

Author: Dennis G. Zill

Publisher: Jones & Bartlett Learning

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Differential Equations

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Chapter 1.1, Problem 19E

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To verify: Whether the given function is an implicit solution of the given differential equation and give the interval definition and graph the explicit solution.

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Chapter 1.1, Problem 18E

Chapter 1.1, Problem 20E

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