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
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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 i...
4 Consider the autonomous differential equation y f(v) a) (3 points) Find all the equilibrium solutions (critical...
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 dy dt = = y(k - y), t> 0, k >...
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.
8. Consider the autonomous DE: y y+1)(y- 2) a) Find and sketch below the equilibrium solutions....
8. Consider the autonomous DE: y y+1)(y- 2) a) Find and sketch below the equilibrium solutions. b) Find the region where the solutions are increasing c) Draw the direction field. d) Sketch three solutions passing respectively through the points (0, 0), (0, 3) and (0, -2) (15 4 2. 0 2 4 2 -2 4
8. Consider the autonomous DE: y y+1)(y- 2) a) Find and sketch below the equilibrium solutions. b) Find the region where the solutions are increasing...
Bifurcation dy Consider the autonomous differential equation =y? - 2y + 8. We will begin by...
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...
(4 points) Write down a first order autonomous differential equation which has equilibrium solutions at y...
(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?
15 pts] Sketch some representative solution curves for the autonomous first order differential equation y'- y(2...
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...
Find an autonomous differential equation with all of the following properties: equilibrium solutions at y=0 and...
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 =
Exercise 4: (5 points) consider the following differential equation 3y - y Let = f(ty) be...
Exercise 4: (5 points) consider the following differential equation 3y - y Let = f(ty) be the right-hand side of the above equation. a. Compute a/ay. b. Determine and sketch the region in the ty-plane where functions. and array are both continuous C. For the initial condition y(0) = 1 (i.e.to = 0, y = 1), would a unique solution of the equation exist? Explain.
Consider the differential equation y' (t) = (y-4)(1 + y). a) Find the solutions that are...
Consider the differential equation y' (t) = (y-4)(1 + y). a) Find the solutions that are constant, for all t2 0 (the equilibrium solutions). b) In what regions are solutions increasing? Decreasing? c) Which initial conditions y(0) = A lead to solutions that are increasing in time? Decreasing? d) Sketch the direction field and verify that it is consistent with parts a through c. a) The solutions are constant for (Type an equation. Use a comma to separate answers as...
consider the autonomous equation 2. Consider the autonomous equation y=-(y2-6y-8) (a) Use the isocline method to ske...
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)...
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 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.
8. Consider the autonomous DE: y y+1)(y- 2) a) Find and sketch below the equilibrium solutions. b) Find the region where the solutions are increasing c) Draw the direction field. d) Sketch three solutions passing respectively through the points (0, 0), (0, 3) and (0, -2) (15 4 2. 0 2 4 2 -2 4
8. Consider the autonomous DE: y y+1)(y- 2) a) Find and sketch below the equilibrium solutions. b) Find the region where the solutions are increasing...
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...
(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?
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...
Exercise 4: (5 points) consider the following differential equation 3y - y Let = f(ty) be the right-hand side of the above equation. a. Compute a/ay. b. Determine and sketch the region in the ty-plane where functions. and array are both continuous C. For the initial condition y(0) = 1 (i.e.to = 0, y = 1), would a unique solution of the equation exist? Explain.
Consider the differential equation y' (t) = (y-4)(1 + y). a) Find the solutions that are constant, for all t2 0 (the equilibrium solutions). b) In what regions are solutions increasing? Decreasing? c) Which initial conditions y(0) = A lead to solutions that are increasing in time? Decreasing? d) Sketch the direction field and verify that it is consistent with parts a through c. a) The solutions are constant for (Type an equation. Use a comma to separate answers as...
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)...
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