Explanation Given information: The given differential equation is y ′ = 5 y − y 2 with the initial condition y ( 0 ) = 1 and y ( 0 ) = 7 . Graph: The graph of the function g ( y ) = 5 y − y 2 on y z − coordinate is shown in the following figure, The zeroes of the graph g ( y ) = 5 y − y 2 are at y = 0 and y = 5 By the property I of differential equations: 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 ′ = 5 y − y 2 has constant solutions at y = 0 and y = 5 The below graph shows a constant solution on t y - coordinate system as follows: Now, begin to sketch the solution satisfying y ( 0 ) = 1 and y ( 0 ) = 7 Locate the initial value of y on the horizontal y - axis in y z graph Locate the initial value of y on the vertical y - axis in t y graph. From the above figure, observe that at y ( 0 ) = 7 , the value of the z − coordinate on y z -graph is negative. Therefore, when the initial condition is y ( 0 ) = 7 , the solution on t y -graph has a negative slope and is decreasing as it leaves the initial point, which is shown in the figure below by a downward arrow. At y ( 0 ) = 1 , the value of the z − coordinate on y z -graph is positive Therefore, when the initial condition is y ( 0 ) = 1 , the solution on t y -graph has a positive slope and is increasing as it leaves the initial point, which is shown in the figure below by an upward arrow. As y will move from y = 1 towards y = 2

12th Edition

ISBN: 9780137590810

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

Publisher: PEARSON

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Chapter 10.5, Problem 23E

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To graph: The solutions of the differential equation y′=5y−y2 with the initial condition y(0)=1 and y(0)=7. Also, indicate the constant solution.

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Chapter 10.5, Problem 22E

Chapter 10.5, Problem 24E

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To graph: The solutions of the differential equation y ′ = 5 y − y 2 with the initial condition y ( 0 ) = 1 and y ( 0 ) = 7 . Also, indicate the constant solution.

Solution Summary: The author illustrates the solution of the differential equation yprime, which has constant solutions on the yz- coordinate system.

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To graph: The solutions of the differential equation y′=5y−y2 with the initial condition y(0)=1 and y(0)=7. Also, indicate the constant solution.

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Chapter 10 Solutions