Consider the nonlinear second-order differential equation where k > 0 is a constant. Answer to the following questio...
Consider the nonlinear second-order differential equation x4 3(x')2 + k2x2 - 1 = 0, _ where k > 0 is a constant. Answer to the following questions. (a) Derive a plane autonomous system from the given equation and find all the critical points (b) Classify(discriminate/categorize) all the critical points into one of the three cat- egories: stable / saddle unstable(not saddle)} (c) Show that there is no periodic solution in a simply connected region {(r, y) R2< 0} R =...
Consider the nonlinear second-order differential equation 4x"+4x'+2(k^2)(x^2)− 1/2 =0, where k > 0 is a constant. Answer to the following questions. (a) Show that there is no periodic solution in a simply connected region R={(x,y) ∈ R2 | x <0}. (Hint: Use the corollary to Theorem 11.5.1>> If symply connected region R either contains no critical points of plane autonomous system or contains a single saddle point, then there are no periodic solutions. ) (b) Derive a plane autonomous system...
4 Consider the autonomous differential equation y f(v) a) (3 points) Find all the equilibrium solutions (critical points). b) (3 points) Use the sign of y f(z) to determine where solutions are increasing / decreasing. Sketch several solution curves in each region determined by the critical points in c) (3 points) the ty-plane. d) (3 points) Classify each equilibrium point as asymptotically stable, unstable, or semi-stable and draw the corresponding phase line. 4 Consider the autonomous differential equation y f(v)...
Without solving explicitly, classify the critical points of the given first-order autonomous differential equation as either asymptotically stable or unstable. All constants are assumed to be positive. (Enter the critical points for each stability category as a comma- separated list. If there are no critical points in a certain category, enter NONE.) dv m = tg - ky dt asymptotically stable VE unstable V mg k х Need Help? Read 1 Talk to a Tutor 2. (-/1 Points] DETAILS ZILLDIFFEQ9...
MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER Consider the following autonomous first-order differential equation. dy = y219 - y2) Find the critical points and phase portrait of the given differential equation. dx 6 3 3 0 0 ol -6 -6 -3 Classify each critical point as asymptotically stable, unstable, or semi-stable. (List the critical points according to their stability, Enter your answers as a comma-separated list. If there are no critical points in a certain category, enter NONE.) asymptotically stable...
Consider the autonomous first-order differential equation y = 10 + 3y – v2 Find the DISTINCT critical points and classify each as (1) AS for Asymptotically Stable, (2) US for Unstable or (3) SS for Semi-Stable. Enter your answer as a comma separated list of pairs consisting on a critical point and its stability type (e.g. your answer might look like (2,AS), (-3,SS), (7,US)) Critical Point and Stability: For the initial value problem y' = 10 + 3y – y,...
Nonlinear differential equations and Bifurcation theory. Given the ordinary differential equation =1+re+ where the parameter r is a given real number. (a) Sketch all the qualitatively different vector fields that occur as r is varied. (b) Show that a saddle-node bifurcation occurs at a critical value of r, to be determined. (c) Sketch the bifurcation diagram of fixed points r* versus r.
Question 3 (20 points) Consider the following differential equation = y(y2 - 4). (a) Find all critical values. (6) Draw the phase diagram to classify each as stable, semi-stable or unstable.
Consider the autonomous differential equation dy dt = = y(k - y), t> 0, k > 0 (i) list the critical points (ii) sketch the phase line and classify the critical points according to their stability (iii) Determine where y is concave up and concave down (iv) sketch several solution curves in the ty-plane.
Consider the following differential equation date = y2(y2 – 4). (a) Find all critical values. (b) Draw the phase diagram to classify each as stable, semi-stable or unstable.