Explanation Given information: The differential equation y ' = y 2 − 6 y + 5 = ( y − 1 ) ( y − 5 ) with initial conditions y ( 0 ) = − 2 , y ( 0 ) = 2 , y ( 0 ) = 4 , y ( 0 ) = 6 and t he graph of y ' = y 2 − 6 y + 5 as shown below: Graph: The differential equation y ' = y 2 − 6 y + 5 is similar to g ( y ) = ( y − 1 ) ( y − 5 ) The graph of the function z = g ( y ) = ( y − 1 ) ( y − 5 ) on a y z -coordinate system is shown in figure (a). From figure (a), the zero of the graph g ( y ) = ( y − 1 ) ( y − 5 ) is at y = 1 and y = 5 By using the following property of autonomous differential equations: 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 2 − 6 y + 5 has constant solutions at y = 1 and y = 5 . The figure (b) shows constant solutions y = 1 and y = 5 on t y - coordinate system as shown below Now, to sketch the solution satisfying y ( 0 ) = − 2 , y ( 0 ) = 2 , y ( 0 ) = 4 , and y ( 0 ) = 6 , locate the initial value of y on horizontal y - axis in figure (a) and vertical y - axis in figure (b) as shown below: To determine whether the solution y ( 0 ) = − 2 , y ( 0 ) = 2 , y ( 0 ) = 4 , and y ( 0 ) = 6 increases or decreases when it leaves the initial points y ( 0 ) on t y -graph and by looking at the y z -graph, it is negative for y ( 0 ) = 2 and y ( 0 ) = 4 and positive for y ( 0 ) = − 2 and y ( 0 ) = 6

12th Edition

ISBN: 9780137590810

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

Publisher: PEARSON

See similar textbooks

Concept explainers

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…

Videos

Question

Chapter 10.5, Problem 12E

To determine

To graph: Thesolutions of the differential equation y' =y2−6y+5=(y −1)(y − 5) with initial conditions y(0) = −2, y(0)=2,y(0) = 4, y(0) = 6 by using the qualitative theory of autonomous differential equations, also include a yz-graph if one is not already provided, then indicate the constant solutions on ty-graph. The graph of z = y2 − 6y+ 5 is as shown below:

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 10.5, Problem 11E

Chapter 10.5, Problem 13E

Students have asked these similar questions

1- √ √ √³ e³/√xdy dx 1 cy² 2- √ √² 3 y³ exy dx dy So 3- √ √sinx y dy dx 4- Jo √² Sy² dx dy

A building that is 205 feet tall casts a shadow of various lengths æ as the day goes by. An angle of elevation is formed by lines from the top and bottom of the building to the tip of the shadow, as de seen in the following figure. Find the rate of change of the angle of elevation when x 278 feet. dx Round to 3 decimal places. Γ X radians per foot

Use the information in the following table to find h' (a) at the given value for a. x|f(x) g(x) f'(x) g(x) 0 0 0 4 3 1 4 4 3 0 2 7 1 2 7 3 3 1 2 9 4 0 4 5 7 h(x) = f(g(x)); a = 0 h' (0) =

Chapter 10 Solutions

Calculus & Its Applications

Ch. 10.1 - Show that any function of the form y=Aet3/3, where...Ch. 10.1 - If the function f(t) is a solution of the...Ch. 10.1 - Prob. 3CYU Ch. 10.1 - Show that the function f(t)=32et212 is a solution...Ch. 10.1 - Show that the function f(t)=t212 is a solution of...Ch. 10.1 - Show that the function f(t)=5e2t satisfies...Ch. 10.1 - Show that the function f(t)=(et+1)1 satisfies...Ch. 10.1 - Prob. 5E Ch. 10.1 - Prob. 6E Ch. 10.1 - Is the constant function f(t)=3 a solution of the...

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

Solution Summary: The author explains the qualitative theory of autonomous differential equations and the graph of y

Concept explainers

Videos

Want to see the full answer?

Chapter 10 Solutions