Explanation Given information: The given differential equation is y ′ = 0.4 y 2 ( 1 − y ) with initial conditions y ( 0 ) = − 1 , y ( 0 ) = 0.1 and y ( 0 ) = 2 . Graph: The graph of a function g ( y ) = 0.4 y 2 ( 1 − y ) on y z coordinate system is shown in the Fig. (a) below: The zeroes of the graph g ( y ) = 0.4 y 2 ( 1 − y ) are at y = 0 and y = 1 . By using the property of an autonomous differential equation: 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 g ( y ) = 0.4 y 2 ( 1 − y ) has constant solutions at y = 0 and y = 1 . The graph of the constant solution on t y - coordinate system is shown in the Fig. (b) below: To sketch of the solution satisfying y ( 0 ) = − 1 , y ( 0 ) = 0.1 and y ( 0 ) = 2 , locate the initial values of y on horizontal y - axis in Fig. (a) and vertical y - axis in Fig. (b). To determine whether the solution y ( 0 ) = − 1 , y ( 0 ) = 0.1 and y ( 0 ) = 2 increases or decreases when it leaves the initial point y ( 0 ) on t y -graph and by looking at the y z -graph is positive for y ( 0 ) = − 1 and negative for y ( 0 ) = 2 , which implies the solution is increasing for y ( 0 ) = − 1 and decreasing for y ( 0 ) = 2 . This is indicated by an arrow at initial point in Fig (b) below: As y will move from y = − 1 towards y = 0 , the z coordinate of points on a graph of g ( y ) becomes less positive, that is g ( y ) becomes less positive, thus y ' = g ( y ) , slope of the solution becomes lesspositive

15th Edition

ISBN: 9780137590896

Author: Larry J. Goldstein; David C. Lay; David I. Schneider; Nakhle H. Asmar; William Edward Tavernetti

Publisher: Pearson Education (US)

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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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System that uses coordinates to uniquely determine the position of points. The most common coordinate system is the Cartesian system, where points are given by distance along a horizontal x-axis and vertical y-axis from the origin. A polar coordinate system locates a point by its direction relative to a reference direction and its distance from a given point. In three dimensions, it leads to cylindrical and spherical coordinates.

Chapter 10, Problem 23RE

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To graph: The solutions of the differential equation y′=0.4y2(1−y) with initial conditions

y(0)=−1, y(0)=0.1 and y(0)=2. Also, indicate the constant solution.

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Chapter 10, Problem 22RE

Chapter 10, Problem 24RE

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To graph: The solutions of the differential equation y ′ = 0.4 y 2 ( 1 − y ) with initial conditions y ( 0 ) = − 1 , y ( 0 ) = 0.1 and y ( 0 ) = 2 . Also, indicate the constant solution.

Solution Summary: The author illustrates the graph of a function g(y)=0.4y

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To graph: The solutions of the differential equation y′=0.4y2(1−y) with initial conditions

y(0)=−1, y(0)=0.1 and y(0)=2. Also, indicate the constant solution.

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Chapter 10 Solutions

To graph: The solutions of the differential equation y ′ = 0.4 y 2 ( 1 − y ) with initial conditions y ( 0 ) = − 1 , y ( 0 ) = 0.1 and y ( 0 ) = 2 . Also, indicate the constant solution.

Solution Summary: The author illustrates the graph of a function g(y)=0.4y

Concept explainers

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To graph: The solutions of the differential equation y′=0.4y2(1−y) with initial conditions y(0)=−1, y(0)=0.1 and y(0)=2. Also, indicate the constant solution.

Want to see the full answer?

Chapter 10 Solutions

To graph: The solutions of the differential equation y′=0.4y2(1−y) with initial conditions

y(0)=−1, y(0)=0.1 and y(0)=2. Also, indicate the constant solution.