Question

2. Consider the initial value problem y'(x) = 2y - y^2 - 1, y(0) = 4 (a) Find any and all equilibrium solutions. (b) Sketch a phase line for this differential equation, and use it to classify each equilibrium as asymptotically stable, unstable, or semi-stable. (c) Based on these conclusions, determine the asymptotic behavior of the above initial value problem. That is, starting from the initial value y(0) = 4, what value will y(x) approach as x \to \infty? (d) Apply Euler's method to numerically approximate the solution of the initial value prob- lem over the domain 0 \le x \le 3 using a step size of h = 0.5. (e) Does the behavior of your numerical solution agree with your earlier asymptotic analysis? If not, what went wrong with the numerical solution process, and how would you suggest fixing it?

          2. Consider the initial value problem
y'(x) = 2y - y^2 - 1, y(0) = 4
(a) Find any and all equilibrium solutions.
(b) Sketch a phase line for this differential equation, and use it to classify each equilibrium
as asymptotically stable, unstable, or semi-stable.
(c) Based on these conclusions, determine the asymptotic behavior of the above initial value
problem. That is, starting from the initial value y(0) = 4, what value will y(x) approach
as x \to \infty?
(d) Apply Euler's method to numerically approximate the solution of the initial value prob-
lem over the domain 0 \le x \le 3 using a step size of h = 0.5.
(e) Does the behavior of your numerical solution agree with your earlier asymptotic analysis?
If not, what went wrong with the numerical solution process, and how would you suggest
fixing it?

2. Consider the initial value problem
y'(x) = 2y - y^2 - 1, y(0) = 4
(a) Find any and all equilibrium solutions.
(b) Sketch a phase line for this differential equation, and use it to classify each equilibrium
as asymptotically stable, unstable, or semi-stable.
(c) Based on these conclusions, determine the asymptotic behavior of the above initial value
problem. That is, starting from the initial value y(0) = 4, what value will y(x) approach
as x →∞?
(d) Apply Euler's method to numerically approximate the solution of the initial value prob-
lem over the domain 0 ≤x ≤3 using a step size of h = 0.5.
(e) Does the behavior of your numerical solution agree with your earlier asymptotic analysis?
If not, what went wrong with the numerical solution process, and how would you suggest
fixing it?

Added by Lidia A.

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