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Differential Equations for Engineers II Page 1 of 6 1. The interface y(x) between air and...
1. The angular displacement e(t) of an undamped pendulum of length l swinging in a vertical...
1. The angular displacement e(t) of an undamped pendulum of length l swinging in a vertical plane under the influence of gravity can be modelled with the nonlinear second order ODE 0"(t) + "$(0(e) – Pef) = = 0, (1) where w2 = { (a) Re-write equation (1) as a system of first-order ODEs by defining (1 = 0(t) and 02 = 't). (b) Find the nullclines and equilibrium points for the system of first-order ODEs. (c) The first order...
Differential Equations for Engineers II Page 3 of 6 3. The interface y(x) between air and...
Differential Equations for Engineers II Page 3 of 6 3. The interface y(x) between air and water in a time-independent open channel flow can be approximated with the second order ODE dạy ta’y = 0, d.r2 >0, (3) 4 marks where the parameter a’ is a measure of the mean speed of the flow. The flow is in the positive x direction (i.e. from left to right). (a) The point x = 0 is an ordinary point of equation (3)....
Consider the non-linear system y-y(1-x-y). (a) Find equations for all of the x- and y-nullclines....
Consider the non-linear system y-y(1-x-y). (a) Find equations for all of the x- and y-nullclines. (b) Find the coordinates of each equilibrium point of the system. (c) Sketch the nullclines in the phase plane. Clearly mark the equilibrium points. Also indicate the direction of flow on the nullclines.
Consider the non-linear system y-y(1-x-y). (a) Find equations for all of the x- and y-nullclines. (b) Find the coordinates of each equilibrium point of the system. (c) Sketch the nullclines in the...
Differential Equations for Engineers II Page 2 of 6 2. Consider the nonhomogeneous ordinary differential equation...
Differential Equations for Engineers II Page 2 of 6 2. Consider the nonhomogeneous ordinary differential equation XY" + 2(x – B)y' + (x – 2B)y = e-1, x > 0, (2) 5 marks where ß > 0 is a given constant. (a) A solution of the associated homogeneous equation is yı = e-*. Use the formula for the method of reduction of order, as described in the lecture notes / record- ings, to find a second solution, y2, of the...
2. (28 marks) This questions is about the following system of equations x = (2-x)(y-1) (a) Find all equilibrium solutio...
2. (28 marks) This questions is about the following system of equations x = (2-x)(y-1) (a) Find all equilibrium solutions and determine their type (e.g., spiral source, saddle) Hint: you should find three equilibria. b) For each of the equilibria you found in part (a), draw a phase portrait showing the behaviour of solutions near that equilibrium. -2 (c) Find the nullclines for the system and sketch them on the answer sheet provided. Show the direction of the vector field...
3. a) Find the solution y(t) of the ordinary differential equation with the initial conditions: (Solve...
3. a) Find the solution y(t) of the ordinary differential equation with the initial conditions: (Solve it only by hand and show your complete work. Do not use a calculator or any symbolic calu lations). [8 marks b) ) Recast our third order ODE into a system of irst order ODEs of the form A.v, where v' = dv/dr = f(v) and v = (y,y,,y")" . You should show all working to find the corresponding matrix A. Do not solve...
3. Consider the system of differential equations = z+9-1 (a) Sketch the r-nullcline, where soluti...
just (A) (B) (C)
3. Consider the system of differential equations = z+9-1 (a) Sketch the r-nullcline, where solutions must travel vertically. Identify the regions (b) On a separate set of axes, sketch the y-nullcline, where solutions must travel horizon in the plane where solutions will move toward the right, and where solutions move toward the right tally. Identify the regions in the plane where solutions will move upward, and where solutions move downward. (c) On a third set of...
Consider the following differential equation system: x' = 16x + 8y y = -24x – 12y...
Consider the following differential equation system: x' = 16x + 8y y = -24x – 12y (a) Find the general solution. (b) Without a computer, sketch a phase diagram that shows four linear solution trajectories and that shows one solution trajectory in each of the four regions between the separatrices. (c) Determine the solution that satisfies x(0) = 1 and y(0) = 0. x(t) = yt) = (d) The point (0,0) is a ... Osaddle point stable node unstable node...
Problem 6. Consider the system: y. and its corresponding vector field: 1. Sketch a number of diff...
Problem 6. Consider the system: y. and its corresponding vector field: 1. Sketch a number of different solution curves on the phase plane. 2. Describe the behavior of the solution that satisfies the initial condition (to, o) (0, 2)
Problem 6. Consider the system: y. and its corresponding vector field: 1. Sketch a number of different solution curves on the phase plane. 2. Describe the behavior of the solution that satisfies the initial condition (to, o) (0, 2)
Consider the differential equation, L[y] = y'' + p(t)y' + q(t)y = 0, (1) whose coefficients...
Consider the differential
equation, L[y] = y'' + p(t)y' + q(t)y = 0, (1) whose coefficients p
and q are continuous on some open interval I. Choose some point t0
in I. Let y1 be the solution of equation (1) that also satisfies
the initial conditions y(t0) = 1, y'(t0) = 0, and let y2 be the
solution of equation (1) that satisfies the initial conditions
y(t0) = 0, y'(t0) = 1. Then y1 and y2 form a fundamental set...
1. The angular displacement e(t) of an undamped pendulum of length l swinging in a vertical plane under the influence of gravity can be modelled with the nonlinear second order ODE 0"(t) + "$(0(e) – Pef) = = 0, (1) where w2 = { (a) Re-write equation (1) as a system of first-order ODEs by defining (1 = 0(t) and 02 = 't). (b) Find the nullclines and equilibrium points for the system of first-order ODEs. (c) The first order...
Differential Equations for Engineers II Page 3 of 6 3. The interface y(x) between air and water in a time-independent open channel flow can be approximated with the second order ODE dạy ta’y = 0, d.r2 >0, (3) 4 marks where the parameter a’ is a measure of the mean speed of the flow. The flow is in the positive x direction (i.e. from left to right). (a) The point x = 0 is an ordinary point of equation (3)....
Consider the non-linear system y-y(1-x-y). (a) Find equations for all of the x- and y-nullclines. (b) Find the coordinates of each equilibrium point of the system. (c) Sketch the nullclines in the phase plane. Clearly mark the equilibrium points. Also indicate the direction of flow on the nullclines.
Consider the non-linear system y-y(1-x-y). (a) Find equations for all of the x- and y-nullclines. (b) Find the coordinates of each equilibrium point of the system. (c) Sketch the nullclines in the...
Differential Equations for Engineers II Page 2 of 6 2. Consider the nonhomogeneous ordinary differential equation XY" + 2(x – B)y' + (x – 2B)y = e-1, x > 0, (2) 5 marks where ß > 0 is a given constant. (a) A solution of the associated homogeneous equation is yı = e-*. Use the formula for the method of reduction of order, as described in the lecture notes / record- ings, to find a second solution, y2, of the...
2. (28 marks) This questions is about the following system of equations x = (2-x)(y-1) (a) Find all equilibrium solutions and determine their type (e.g., spiral source, saddle) Hint: you should find three equilibria. b) For each of the equilibria you found in part (a), draw a phase portrait showing the behaviour of solutions near that equilibrium. -2 (c) Find the nullclines for the system and sketch them on the answer sheet provided. Show the direction of the vector field...
3. a) Find the solution y(t) of the ordinary differential equation with the initial conditions: (Solve it only by hand and show your complete work. Do not use a calculator or any symbolic calu lations). [8 marks b) ) Recast our third order ODE into a system of irst order ODEs of the form A.v, where v' = dv/dr = f(v) and v = (y,y,,y")" . You should show all working to find the corresponding matrix A. Do not solve...
just (A) (B) (C)
3. Consider the system of differential equations = z+9-1 (a) Sketch the r-nullcline, where solutions must travel vertically. Identify the regions (b) On a separate set of axes, sketch the y-nullcline, where solutions must travel horizon in the plane where solutions will move toward the right, and where solutions move toward the right tally. Identify the regions in the plane where solutions will move upward, and where solutions move downward. (c) On a third set of...
Consider the following differential equation system: x' = 16x + 8y y = -24x – 12y (a) Find the general solution. (b) Without a computer, sketch a phase diagram that shows four linear solution trajectories and that shows one solution trajectory in each of the four regions between the separatrices. (c) Determine the solution that satisfies x(0) = 1 and y(0) = 0. x(t) = yt) = (d) The point (0,0) is a ... Osaddle point stable node unstable node...
Problem 6. Consider the system: y. and its corresponding vector field: 1. Sketch a number of different solution curves on the phase plane. 2. Describe the behavior of the solution that satisfies the initial condition (to, o) (0, 2)
Problem 6. Consider the system: y. and its corresponding vector field: 1. Sketch a number of different solution curves on the phase plane. 2. Describe the behavior of the solution that satisfies the initial condition (to, o) (0, 2)
Consider the differential
equation, L[y] = y'' + p(t)y' + q(t)y = 0, (1) whose coefficients p
and q are continuous on some open interval I. Choose some point t0
in I. Let y1 be the solution of equation (1) that also satisfies
the initial conditions y(t0) = 1, y'(t0) = 0, and let y2 be the
solution of equation (1) that satisfies the initial conditions
y(t0) = 0, y'(t0) = 1. Then y1 and y2 form a fundamental set...
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