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3. A ODE is called autonomous if it contains no independent variables. That is y f...
Consider an autonomous ODE y = f(y) where f(y) = y2 - 1 A. (1 pt)...
Consider an autonomous ODE y = f(y) where f(y) = y2 - 1 A. (1 pt) Draw the graph of function f(y) in the y,y plane and specify the points y where f(y) is singular (that is, f(y) takes an infinite value). B. (1 pt) Finf the equilibrium solution and determine its type: stable, un- stable or semi-stable. Indicate on which side it attracts/repels nearby solutions. C. (2 pt) By separating y and t, specify the general solution y=yt,C) t+C...
1. (25 pts) An autonomous differential equation has an unstable equilibrium solution at y = -1,...
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?
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...
3.) (1 point) Revisit our example from class and estimate y(O) for the ODE: y'= y...
3.) (1 point) Revisit our example from class and estimate y(O) for the ODE: y'= y + x2 y(-1) = -1, using Euler's Method and an h = 0.25 (4 steps) Also plot the direction field for this ODE from -2<x<2 and -2<y<2, only use integer values of x and y for this plot. Draw a line of the particular solution based on your estimate of y(0)
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)...
If X and Y are independent random variables with f(x)= x/2 for 0<x<2 and g(y)=...
If X and Y are independent random variables with f(x)= x/2 for 0<x<2 and g(y)= 2y 0<y<1 Compute: E(XY)
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)...
2. The following ODE model (for the Duffing oscillator) describes the motion of a damped spring d...
2. The following ODE model (for the Duffing oscillator) describes the motion of a damped spring driven by a periodic force: r(0) = zo (a) Rewrite the second order non-autonomous system in one independent variable above as an autonomous system in three independent variables: x, y and r, where: y-r ano T 1, with T(0)-0 (b) Fix the parameter values of α = 1, β-0, δ 0.05, w-1. Additionally, fix the initial conditions 2(0)-10, z'(0) . For the values of...
The phase plot for an ODE dy dx =f(y) dydx=f(y) is shown below. 4 3 2 1 2 1 1 1 1 2 3 (a) Which of these could be a plot of solutions y vs x corresponding to this ODE? 9 2 B. A. 2 2 3 C. D. You ca...
The phase plot for an ODE dy dx =f(y) dydx=f(y) is shown
below.
4 3 2 1 2 1 1 1 1 2 3 (a) Which of these could be a plot of solutions y vs x corresponding to this ODE? 9 2 B. A. 2 2 3 C. D. You can click the graphs above to enlarge them. OA. A ов, в OC. C OD. D E which is choose (b) The smallest equilibrium of this ODE is y-...
3. Continuous dynamical systems - Dimension 2 (a) Suppose the ODE system describes a continuous dynamical...
3. Continuous dynamical systems - Dimension 2 (a) Suppose the ODE system describes a continuous dynamical system in two dimensions (here f: R2 + R and g: R² R are two functions with smooth partial derivatives). Draw the corresponding vector field in the case that f(x,y) = x2 - y2 8(x, y) = x+y+1 and argue that (x,y) = R2 such that f(x,y) = g(x,y) = 0 are fixed points of the dynamical system above.
Consider an autonomous ODE y = f(y) where f(y) = y2 - 1 A. (1 pt) Draw the graph of function f(y) in the y,y plane and specify the points y where f(y) is singular (that is, f(y) takes an infinite value). B. (1 pt) Finf the equilibrium solution and determine its type: stable, un- stable or semi-stable. Indicate on which side it attracts/repels nearby solutions. C. (2 pt) By separating y and t, specify the general solution y=yt,C) t+C...
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?
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...
3.) (1 point) Revisit our example from class and estimate y(O) for the ODE: y'= y + x2 y(-1) = -1, using Euler's Method and an h = 0.25 (4 steps) Also plot the direction field for this ODE from -2<x<2 and -2<y<2, only use integer values of x and y for this plot. Draw a line of the particular solution based on your estimate of y(0)
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)...
2. The following ODE model (for the Duffing oscillator) describes the motion of a damped spring driven by a periodic force: r(0) = zo (a) Rewrite the second order non-autonomous system in one independent variable above as an autonomous system in three independent variables: x, y and r, where: y-r ano T 1, with T(0)-0 (b) Fix the parameter values of α = 1, β-0, δ 0.05, w-1. Additionally, fix the initial conditions 2(0)-10, z'(0) . For the values of...
The phase plot for an ODE dy dx =f(y) dydx=f(y) is shown
below.
4 3 2 1 2 1 1 1 1 2 3 (a) Which of these could be a plot of solutions y vs x corresponding to this ODE? 9 2 B. A. 2 2 3 C. D. You can click the graphs above to enlarge them. OA. A ов, в OC. C OD. D E which is choose (b) The smallest equilibrium of this ODE is y-...
3. Continuous dynamical systems - Dimension 2 (a) Suppose the ODE system describes a continuous dynamical system in two dimensions (here f: R2 + R and g: R² R are two functions with smooth partial derivatives). Draw the corresponding vector field in the case that f(x,y) = x2 - y2 8(x, y) = x+y+1 and argue that (x,y) = R2 such that f(x,y) = g(x,y) = 0 are fixed points of the dynamical system above.
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