Explanation Given information: The differential equation y ' = y − 1 4 y 2 with initial conditions y ( 0 ) = − 1 , y ( 0 ) = 1 Graph: The zeroes of the graph g ( y ) = y − 1 4 y 2 are at y = 0 and y = 4 . By using the property of an autonomous differential equation: For each zero of g ( y ) , there is a constant solution of the differential equation. Specifically, if g ( c ) = 0 , the constant function y = c is a solution. Thus, the differential equation y ' = y − 1 4 y 2 has constant solutions at y = 0 and at y = 4 . The graph of the function z = g ( y ) = y − 1 4 y 2 on a y z -coordinate system and the graph of the constant solutions on t y - coordinate system is shown in the Fig. (a) and Fig. (b) below: To sketch of the solution satisfying y ( 0 ) = − 1 and y ( 0 ) = 1 , locate the initial value of y on the horizontal y - axis in Fig. (a) and vertical y - axis in Fig. (b).

Pearson eText for Calculus & Its Applications -- Instant Access (Pearson+)

15th Edition

ISBN: 9780137590728

Author: Larry Goldstein, David Lay

Publisher: PEARSON+

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Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differentia…

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Chapter 10.5, Problem 28E

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To graph: The differential equation y' = y − 14y2 with initial conditions y(0) = −1, y(0) = 1

by using the qualitative theory of autonomous differential equation; also, include the yz-graph, if one is not already provided and indicate the constant solution on ty-graph.

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Chapter 10.5, Problem 27E

Chapter 10.5, Problem 29E

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To graph: The differential equation y ' = y − 1 4 y 2 with initial conditions y ( 0 ) = − 1 , y ( 0 ) = 1 by using the qualitative theory of autonomous differential equation; also, include the y z -graph, if one is not already provided and indicate the constant solution on t y -graph.

Solution Summary: The author illustrates the graph of the differential equation y's value by using the qualitative theory of autonomous equation.

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To graph: The differential equation y' = y − 14y2 with initial conditions y(0) = −1, y(0) = 1

by using the qualitative theory of autonomous differential equation; also, include the yz-graph, if one is not already provided and indicate the constant solution on ty-graph.

Want to see the full answer?

Chapter 10 Solutions

To graph: The differential equation y ' = y − 1 4 y 2 with initial conditions y ( 0 ) = − 1 , y ( 0 ) = 1 by using the qualitative theory of autonomous differential equation; also, include the y z -graph, if one is not already provided and indicate the constant solution on t y -graph.

Solution Summary: The author illustrates the graph of the differential equation y's value by using the qualitative theory of autonomous equation.

Concept explainers

Videos

To graph: The differential equation y' = y − 14y2 with initial conditions y(0) = −1, y(0) = 1 by using the qualitative theory of autonomous differential equation; also, include the yz-graph, if one is not already provided and indicate the constant solution on ty-graph.

Want to see the full answer?

Chapter 10 Solutions

To graph: The differential equation y' = y − 14y2 with initial conditions y(0) = −1, y(0) = 1

by using the qualitative theory of autonomous differential equation; also, include the yz-graph, if one is not already provided and indicate the constant solution on ty-graph.