For each of the following differential equations of the form # = f(y), sketch a graph of f(y) versus y, determine the equilibrium (critical) points, and classify each one as asymptotically stable, unstable, or semistable. Draw a phase line, and sketch enough integral curves in the ty-plane so that the behavior for any initial condition y(0) = y0 is shown. %3D (a) # = ay + by², a > 0, b > 0, yo 2 0 (b) = ev – 1, -∞ < Yo < ∞ (c) = y(1 – y²), -∞ < yo < o∞ %3D %3D %3D

For each of the following differential equations of the form # = f(y), sketch a graph of f(y) versus y, determine the equilibrium (critical) points, and classify each one as asymptotically stable, unstable, or semistable. Draw a phase line, and sketch enough integral curves in the ty-plane so that the behavior for any initial condition y(0) = y0 is shown. %3D (a) # = ay + by², a > 0, b > 0, yo 2 0 (b) = ev – 1, -∞ < Yo < ∞ (c) = y(1 – y²), -∞ < yo < o∞ %3D %3D %3D

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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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 differential equations.

Question

For each of the following differential equations of the form * = f(y), sketch a graph
of f(y) versus y, determine the equilibrium (critical) points, and classify each one as
asymptotically stable, unstable, or semistable. Draw a phase line, and sketch enough
integral curves in the ty-plane so that the behavior for any initial condition y(0) = yo
is shown.
(a) = ay + by², a > 0, b > 0, yo > 0
dt
(b) 4 = ev – 1, -0 < Yo < ∞
(c) = y(1 – y²), –∞ < yo <∞

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