15 pts] Sketch some representative solution curves for the autonomous first order differential equation y'- y(2...
Bifurcation dy Consider the autonomous differential equation =y? - 2y + 8. We will begin by examining dt the equilibrium solutions of the equation for various values of the parameter 8 1. Find the equilibrium solutions of the equation for 8 = -4,-2, 0, 2, 4 and make a sketch of the phase line for each value. Determine the stability of each equilibria. 2. Use a computer or some other means to sketch some solution curves for each value of...
sketch the slope field and some representative solution curves for the given differential equation. For Problems 22–29, sketch the slope field and some repre- sentative solution curves for the given differential equation. 22. y' = 4x.
DO HAND CALCULATIONS. SHOW ALL STEPS 1. Slope Fields For the given differential equations sketch the slope fields and some of the isoclines. Then sketch some of the solution curves and verify your answer by solving the differential equation. a) dy-2 dx y 1. Slope Fields For the given differential equations sketch the slope fields and some of the isoclines. Then sketch some of the solution curves and verify your answer by solving the differential equation. a) dy-2 dx y
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 autonomous differential equation y = f(y) = y4-4 уг = y"(y-2) (y+2). a) (3 points) Find all the equilibrium solutions (critical points). f(y) to determine where solutions are increasing / decreasing. Use the sign of y' e) (3 points) Sketch several solution curves in each region determined by the critical poins in the ty-plane Consider the autonomous differential equation y = f(y) = y4-4 уг = y"(y-2) (y+2). a) (3 points) Find all the equilibrium solutions (critical points)....
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)...
Find an autonomous differential equation with all of the following properties: equilibrium solutions at y=0 and y=3, y' > 0 for 0<y<3 and y' < 0 for -inf < y < 0 and 3 < y < inf dy/dx =
An autonomous system of two first order differential equations can be written as: A third order explicit Runge-Kutta scheme for an autonomous system of two first order equations is Consider the following second order differential equation, Use the Runge-Kutta scheme to find an approximate solutions of the second order differential equation, at t = 1.2, if the step size h = 0.1. Maintain at least eight decimal digit accuracy throughout all your calculations. You may express your answer as a...
consider the autonomous equation 2. Consider the autonomous equation y=-(y2-6y-8) (a) Use the isocline method to sketch a direction field for the equation (b) Sketch the solution curves corresponding to the following intitial conditions: (1) y(0) 1 (2) y(0) =3 (3) y(0)=5 (4) 3y(0) 2 (5) y(0) = 4 (c) What are equilibrium solutions, and classify its equilibrium them as: sink (stable), source, node. (d) What is limy(t) if y(0) = 6? too 2. Consider the autonomous equation y=-(y2-6y-8) (a)...
1. (25 pts) An autonomous differential equation has an unstable equilibrium solution at y = -1, a semi-stable equilibrium solution at y = 0, and a stable equilibrium at y = 5/2. a. Sketch the slope field for the system. b. Propose a first order differential equation (use x as the independent variable) that meets the description above. c. What solution method(s) can be used to solve this system?