Explanation The tangent will be vertical when the slope ( y ′ ) is undefined. Consider the given differential equation. d y d x = y ( y 3 − 2 x 3 ) x ( 2 y 3 − x 3 ) For slope to be undefined here, x ( 2 y 3 − x 3 ) should be zero as follows, x ( 2 y 3 − x 3 ) = 0 Simplify further for x . x = 0 And, 2 y 3 − x 3 = 0 2 y 3 = x 3 y 3 = x 3 2 (1) Consider the given family of one parameter. x 3 + y 3 = 3 x y (2) Substitute x 3 2 for y 3 in Equation (2). x 3 + x 3 2 = 3 x ( x 2 1 3 ) 3 2 x 3 = 3 x 2 2 1 3 x 3 = 2 2 3 x 2 x 2 ( x − 2 2 3 ) = 0 Simplify further for the values of x

Bundle: Differential Equations with Boundary-Value Problems, 9th + WebAssign Printed Access Card for Zill's Differential Equations with Boundary-Value Problems, 9th Edition, Single-Term

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ISBN: 9781337604918

Author: Dennis G. Zill

Publisher: Cengage Learning

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Chapter 1.1, Problem 52E

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To discuss: How can a differential equation help in finding points on the graph.

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Chapter 1.1, Problem 51E

Chapter 1.1, Problem 53E

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How can a differential equation help in finding points on the graph.

Solution Summary: The author explains how a differential equation helps in finding points on the graph. Vertical tangent lines are at points (0,0) and

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