(4 points) Write down a first order autonomous differential equation which has equilibrium solutions at y...
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)...
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)....
15 pts] Sketch some representative solution curves for the autonomous first order differential equation y'- y(2-y) (1 -y). Find all equilibrium solutions, label all pertinent coordinates. Note: An Autonomous equation means that dy/dt does not depend on time t. Hint: Follow the method demonstrated in Example 1.3.6 (p.28). The hand-draw slop field is optional and not necessary. This method gives a qualitative analysis for the future of all possible solutions without solving the equation quantitatively 15 pts] Sketch some representative...
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?
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 =
(17 points) A 9th order, linear, homogeneous, constant coefficient differential equation has a characteristic equation which factors as follows. (P2 - 2r+2) rtr + 3) = 0 Write the nine fundamental solutions to the differential equation. y y2 > Y4= Ys = Y y = 19 = (You can enter your answers in any order.)
Consider the following autonomous first-order differential equation. Find the critical points and phase portrait of the given differential equation. 0 Consider the following autonomous first-order differential equation. Find the critical points and phase portrait of the given differential equation. 0
6. Consider the autonomous differential equation (a) Find all of its equilibrium solutions. (b) Classify the stability of each equilibrium solution. Justify your answer. (c) If y(t) is a solution that satisfies y(-1) =-4, what is y(0)? Without solving the equation, briefly explain your conclusion. (d) If y() is a solution that satisis y(3) -3, then what is lim y(t)? 6. Consider the autonomous differential equation (a) Find all of its equilibrium solutions. (b) Classify the stability of each equilibrium...
Q1. Below is the bifurcation diagram for a first-order differential equation. y- 2 k=0 k-4 (a) Write down a first-order differential equation that would have this bifur- cation diagram. tative behaviour of solutions occurs. lution if (b) Find bifurcation values of k, i.e., values of k where a change in the quali- uit so. Q1. Below is the bifurcation diagram for a first-order differential equation. y- 2 k=0 k-4 (a) Write down a first-order differential equation that would have this...
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...