Find an autonomous differential equation with all of the following properties:
dy/dx =
Find an autonomous differential equation with all of the following properties: equilibrium solutions at y=0 and...
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)...
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...
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...
(4 points) Write down a first order autonomous differential equation which has equilibrium solutions at y = 1, y = 2 and y = 3. Is the differential equation linear?
Find the stable equilibrium solution of the following differential equation: + y - 1 = e2( y − 1). The stable equilibrium solution is y = Check Find the general solution to the differential equation: x + y - x115 = 0. Answer: y(x) = Check Solve the initial-value problem: dy = e ** - y, yO= dx Answer: y(x) = Check
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...
Find all equilibrium solutions for the following autonomous equation, and determine the stability of each equilibrium. (Enter your answers from smallest to largest.) dt x which is semistable X which is stableX X which is unstable x
e critical points for the autonomous equation y'=y (ay) e whether they lead to equilibrium solubions which are shable, unstable or semesbable equilíbria. A solue g =- Jy y (0)=50 X+goo o solve y + y = x 0 Prove that the equation cosy dx - (x sing-y dy=0 s excent and then solve. 2 Prove that the ogrubion (easing) dx+cos ydy=0 is nob excent and then find an anbegrubina factor which will make it exact. Prove that exy dx...
3. Find all a,b such that all solutions of the differential equation y+ ay+by 0 converge to zero asgoes to