Explanation Given information: y ' = y 3 − 9 y or y ' = y ( y 2 − 9 ) with initial conditions y ( 0 ) = − 4 , y ( 0 ) = − 1 , y ( 0 ) = 2 , y ( 0 ) = 4 . The graph of y ' = y 3 − 9 y : Graph: y ' = y ( y 2 − 9 ) is similar to g ( y ) = y ( y 2 − 9 ) . On a y z -coordinate system to draw the graph of the function z = g ( y ) = y ( y 2 − 9 ) Observe the graph of g ( y ) = y ( y 2 − 9 ) in fig (a). The zero of the graph g ( y ) = y ( y 2 − 9 ) is at y = − 3 , y = 0 and y = 3 By the property of differential equation: Corresponding to 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 ( y 2 − 9 ) has constant solution at y = − 3 , y = 0 and y = 3 . The graph (b) shows constant solution on t y - coordinate system as follows: Now, begin to sketch the solution satisfying y ( 0 ) = − 4 , y ( 0 ) = − 1 , y ( 0 ) = 2 , y ( 0 ) = 4 . Locate the initial value of y on horizontal y - axis in graph (a) and vertical y - axis in graph (b) To determine whether the solution y ( 0 ) = − 4 , y ( 0 ) = − 1 , y ( 0 ) = 2 , y ( 0 ) = 4 increases or decreases, when it leaves the initial point y ( 0 ) on t y -graph and by looking at the y z -graph, it is negative for y ( 0 ) = − 4 and y ( 0 ) = 2 and positive for y ( 0 ) = − 1 and y ( 0 ) = 4 . Which implies the solution is decreasing for both y ( 0 ) = − 4 and y ( 0 ) = 2 and increasing for y ( 0 ) = − 1 and y ( 0 ) = 4

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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Chapter 10.5, Problem 13E

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To graph: y' = y3−9y or y' =y(y2−9) with initial conditions y(0) = −4, y(0) = −1, y(0) = 2,y(0)=4 using the qualitative theory of autonomous differential equations. Include a yz-graph if one is not already provided and indicate the constant solutions on ty-graph, where the graph of y' = y3−9y is given bellow:

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Chapter 10.5, Problem 12E

Chapter 10.5, Problem 14E

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Solution Summary: The author explains how to draw a graph of y's --sequences using the qualitative theory of autonomous differential equations.

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