10. Consider the following autonomous DE: y' = y² (4- y²). (a) Find the equilibrium solutions and classify them as stable, semistable or unstable. Sketch a direction field and several graphs of solutions in the ty-plane. Also graph y' versus y and the phase line. (b) Find a formula for y" and use this to determine the concavity of solutions for certain values of y.
10. Consider the following autonomous DE: y' = y² (4- y²). (a) Find the equilibrium solutions and classify them as stable, semistable or unstable. Sketch a direction field and several graphs of solutions in the ty-plane. Also graph y' versus y and the phase line. (b) Find a formula for y" and use this to determine the concavity of solutions for certain values of y.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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please solve it on paper
the 2nd pic is from the answer key
Transcribed Image Text:a. Equilibrium solutions: y = −2 (unstable), y = 0 (semistable), y = 2 (stable)
b. y' = 4y³ (2+y)(2 −y)(√2+y)(√2-y)
-
Concave down for −∞ < y < −2, -√√2 < y < 0,√2 < y < 2
Concave up for -2 < y < -√2,0 < y < √2,2 < y < ∞
Transcribed Image Text:10. Consider the following autonomous DE:
y' = y² (4- y²).
(a) Find the equilibrium solutions and classify them as stable, semistable or unstable. Sketch a direction
field and several graphs of solutions in the ty-plane. Also graph y' versus y and the phase line.
(b) Find a formula for y" and use this to determine the concavity of solutions for certain values of y.
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