Explanation Calculation: Consider the differential equation of the given form. d y d x = f ( y ) y ( x ) is the solution of the autonomous differential equation and the equation in a sub-region R is bounded by two consecutive critical points c 1 < c 2 . Assume f ( y ) and f ′ ( y ) for the differential equation are continuous. The point y = c 1 within R such that f ( c 1 ) = 0 , if f ( y ) changes sign within a sub region R of the phase portrait. So, there will be an equilibrium solution y = c 1 within sub-region R . lim x → ∞ y ( x ) = c 2 lim x → ∞ y ′ ( x ) = 0 If this new critical point is between it will be a contradiction of consecutive critical points. A solution y(x) of d y d x = f ( y ) cannot have relative extrema and hence must be monotone. Since, y ′ ( x ) = f ( y ) > 0 , y ( x ) is monotone increasing, and since y ( x ) is bounded above by c 2

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

9th Edition

ISBN: 9781305965799

Author: Dennis G. Zill

Publisher: Cengage Learning

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Chapter 2.1, Problem 34E

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The reason for the number L<c2 cannot exist for the given limit.

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

Ch. 2.1 - In Problems 14 reproduce the given...Ch. 2.1 - In Problems 14 reproduce the given...Ch. 2.1 - dydx=1xy (a) y(0) = 0 (b) y(1) = 0 (c) y(2) = 2...Ch. 2.1 - In Problems 14 reproduce the given...Ch. 2.1 - Prob. 5E Ch. 2.1 - Prob. 6E Ch. 2.1 - Prob. 7E Ch. 2.1 - Prob. 8E Ch. 2.1 - Prob. 9E Ch. 2.1 - Prob. 10E

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The reason for the number L < c 2 cannot exist for the given limit.

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The reason for the number L &lt; c 2 cannot exist for the given limit.

Solution Summary: The author explains that the number Lc_2 cannot exist for the given limit.

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The reason for the number L<c2 cannot exist for the given limit.

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The reason for the number L < c 2 cannot exist for the given limit.