Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
3. dy/dt = e) -1, -∞

3. dy/dt = e) -1, -∞

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
See similar textbooks
icon
Related questions
Question

3

come tion rom mate ing ond rsus e of (18) Problems Problems I through 4 involve equations of the form dy/dt = f(y). In each problem sketch the graph of f(y) versus y, determine the critical (equilibrium) points, and classify each one as asymptotically stable or unstable. Draw the phase line, and sketch several graphs of solutions in the ry-plane. G 1. dy/dt = ay+by², a > 0, b>0, -∞<yo <∞ dy/dt = y(y - 1)(y-2), Yo ≥ 0 G 2. 3. dy/dt = e" - 1, -∞ 108 4. dy/dt = e) -1,00<yo <∞ G G 5. Semistable Equilibrium Solutions. Sometimes a constant equilibrium solution has the property that solutions lying on one side of the equilibrium solution tend to approach it, whereas solutions lying on the other side depart from it (see Figure 2.5.9). In this case the equilibrium solution is said to be semistable. a. Consider the equation dy/dt = k(1 - y)², (19) where k is a positive constant. Show that y = 1 is the only critical point, with the corresponding equilibrium solution (t) = 1. Gb. Sketch f(y) versus y. Show that y is increasing as a function of t for y< 1 and also for y> 1. The phase line has upward-pointing arrows both below and above y = 1. Thus solutions below the equilibrium solution approach it, and those above it grow farther away. Therefore, (t) = 1 is semistable. c. Solve equation (19) subject to the initial condition y(0) = yo and confirm the conclusions reached in part b. У4 k o(t) = k y k o(t) = k 10. C (11) of 11. In equatio 12. C in the s finding equatic 13. C inflect thresh 14. C critica solutio if f'( 15. dy/d G grov whe
Transcribed Image Text:come tion rom mate ing ond rsus e of (18) Problems Problems I through 4 involve equations of the form dy/dt = f(y). In each problem sketch the graph of f(y) versus y, determine the critical (equilibrium) points, and classify each one as asymptotically stable or unstable. Draw the phase line, and sketch several graphs of solutions in the ry-plane. G 1. dy/dt = ay+by², a > 0, b>0, -∞<yo <∞ dy/dt = y(y - 1)(y-2), Yo ≥ 0 G 2. 3. dy/dt = e" - 1, -∞ 108 4. dy/dt = e) -1,00<yo <∞ G G 5. Semistable Equilibrium Solutions. Sometimes a constant equilibrium solution has the property that solutions lying on one side of the equilibrium solution tend to approach it, whereas solutions lying on the other side depart from it (see Figure 2.5.9). In this case the equilibrium solution is said to be semistable. a. Consider the equation dy/dt = k(1 - y)², (19) where k is a positive constant. Show that y = 1 is the only critical point, with the corresponding equilibrium solution (t) = 1. Gb. Sketch f(y) versus y. Show that y is increasing as a function of t for y< 1 and also for y> 1. The phase line has upward-pointing arrows both below and above y = 1. Thus solutions below the equilibrium solution approach it, and those above it grow farther away. Therefore, (t) = 1 is semistable. c. Solve equation (19) subject to the initial condition y(0) = yo and confirm the conclusions reached in part b. У4 k o(t) = k y k o(t) = k 10. C (11) of 11. In equatio 12. C in the s finding equatic 13. C inflect thresh 14. C critica solutio if f'( 15. dy/d G grov whe
Expert Solution
steps

Step by step

Solved in 2 steps with 3 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS