1. Consider the system x˙ = xy2 + x 2 y + x 3 , y˙ = −x 3 + y 3 . Is (0, 0) attracting/stable or unstable? Give a proof.
Homework Answers
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...
6. Consider the following system x' = (^2 3)x. (a) Show that the eigenvalues are given...
6. Consider the following system x' = (^2 3)x. (a) Show that the eigenvalues are given by y + y2 - 16 2 (b) Give the interval(s) of y where the eigenvalues are real. (c) Give the interval(s) where the real eigenvalues give an asymptotically stable node. (d) Give the interval(s) where the real eigenvalues give an unstable node. (e) Give the interval(s) where the complex eigenvalues give an asymptotically stable spiral. (f) Give the interval(s) where the complex eigenvalues...
Problem 3: Study the planar system x' = (y – 1)(y – 2), - . Find...
Problem 3: Study the planar system x' = (y – 1)(y – 2), - . Find and sketch the nullclines and give the vectorfield on the nullelines. Also find all steady states and perform a linear stability analysis for each of them to find whether they are stable or unstable and give the type (saddle, node, focus).
The joint pdf of X and Y is f(x,y)= { (1 + xy2) 0 < x...
The joint pdf of X and Y is f(x,y)= { (1 + xy2) 0 < x < y < 1 otherwise. 0 Find E(X Y = y) 5y2 6 543 27 y2 + + cola 2 3y+2y4 3(73+2)
For each of the following systems, find a > 0 and b > 0 such that L(x, y) = ax^2 + by^2 obe...
For each of the following systems, find a > 0 and b > 0
such that L(x, y) = ax^2 + by^2 obeys d/dt(L) not = 0 whenever (x,
y) 6= (0, 0). (This makes L a Liapounov function.) State whether
the origin is a stable or unstable equilibrium in each case.
(a) x' = −x^3 + 7xy^2 , y' = −3x^2y + y^3 .
(b) x' = x^3 − y^3 , y' = 3xy^2 + 4x^2 y + 5y^3...
Consider the following. x = 8x + y y' - 2x + 6y. X(O) = (-1,2)...
Consider the following. x = 8x + y y' - 2x + 6y. X(O) = (-1,2) (a) Find the general solution (x(t), y(t) = Determine whether there are periodic solutions. (If there are periodic solutions, enter the period. If not, enter NONE.) NONE (b) Find the solution satisfying the given initial condition (x(6), y(t)) - (c) With the aid of a calculator or a CAS graph the solution in part (b) and indicate the direction in which the curve is...
Consider the plane autonomous system 4) 2 X'=AX with A (a) Find two linearly independent real solutions of the syst...
Consider the plane autonomous system 4) 2 X'=AX with A (a) Find two linearly independent real solutions of the system (b) Classify the stability (stable or unstable) and the type (center, node, saddle, or spiral) of the critical point (0,0). (c) Plot the phase portrait of the system containing a trajectory with direction as t-oo whose initial value is X(0) (0,6)7 and any other trajectory with direc- tion. (You do not need to draw solution curves explicitly.)
Consider the plane...
Consider an autonomous system , = (1 + c)x + cy where c is a real constant. (a) Calculate the trace T and the determina...
Consider an autonomous system , = (1 + c)x + cy where c is a real constant. (a) Calculate the trace T and the determinant of the coefficient matrix c+1 c (b) For each following cases of c, classify the stability (stable or unstable) and the type (center, node, saddle, or spiral) of the critical point (0,0). Note that if a critical point is a center, it is stablhe. (4) c=흘 (2) c=-2 (1) c=-1 (3) c=-8
Consider an autonomous...
Consider an LTI system whose input x[n] and output y[n] are related by the difference equation...
Consider an LTI system whose input x[n] and output y[n] are related by the difference equation y[n – 1] + 3 y[n] + $y[n + 1] = x[n]. Determine the three possible choices for the impulse response that makes this system 1) causal, 2) two-sided and 3) anti-causal. Then for each case, determine if the system is stable or not. Causality Impulse Response Stability Causal Unstable v two-sided Unstable anti-Causal Unstable y In your answers, enter z(n) for a discrete-time...
3. Consider an autonomous system z'=cx +2y where c is a real constant. 2 (a) Calculate the trace T and the determin...
3. Consider an autonomous system z'=cx +2y where c is a real constant. 2 (a) Calculate the trace T and the determinant A of the coefficient matrix -2 1 (b) For each following cases of c, classify the stability (stable or unstable) and the type (center, node, saddle, or spiral) of the critical point (0,0). Note that if a critical point is a center, it is stable. (1) c--5 (2) c--3 (3) c1 (4) c 6
3. Consider an autonomous...
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...
6. Consider the following system x' = (^2 3)x. (a) Show that the eigenvalues are given by y + y2 - 16 2 (b) Give the interval(s) of y where the eigenvalues are real. (c) Give the interval(s) where the real eigenvalues give an asymptotically stable node. (d) Give the interval(s) where the real eigenvalues give an unstable node. (e) Give the interval(s) where the complex eigenvalues give an asymptotically stable spiral. (f) Give the interval(s) where the complex eigenvalues...
Problem 3: Study the planar system x' = (y – 1)(y – 2), - . Find and sketch the nullclines and give the vectorfield on the nullelines. Also find all steady states and perform a linear stability analysis for each of them to find whether they are stable or unstable and give the type (saddle, node, focus).
The joint pdf of X and Y is f(x,y)= { (1 + xy2) 0 < x < y < 1 otherwise. 0 Find E(X Y = y) 5y2 6 543 27 y2 + + cola 2 3y+2y4 3(73+2)
For each of the following systems, find a > 0 and b > 0
such that L(x, y) = ax^2 + by^2 obeys d/dt(L) not = 0 whenever (x,
y) 6= (0, 0). (This makes L a Liapounov function.) State whether
the origin is a stable or unstable equilibrium in each case.
(a) x' = −x^3 + 7xy^2 , y' = −3x^2y + y^3 .
(b) x' = x^3 − y^3 , y' = 3xy^2 + 4x^2 y + 5y^3...
Consider the following. x = 8x + y y' - 2x + 6y. X(O) = (-1,2) (a) Find the general solution (x(t), y(t) = Determine whether there are periodic solutions. (If there are periodic solutions, enter the period. If not, enter NONE.) NONE (b) Find the solution satisfying the given initial condition (x(6), y(t)) - (c) With the aid of a calculator or a CAS graph the solution in part (b) and indicate the direction in which the curve is...
Consider the plane autonomous system 4) 2 X'=AX with A (a) Find two linearly independent real solutions of the system (b) Classify the stability (stable or unstable) and the type (center, node, saddle, or spiral) of the critical point (0,0). (c) Plot the phase portrait of the system containing a trajectory with direction as t-oo whose initial value is X(0) (0,6)7 and any other trajectory with direc- tion. (You do not need to draw solution curves explicitly.)
Consider the plane...
Consider an autonomous system , = (1 + c)x + cy where c is a real constant. (a) Calculate the trace T and the determinant of the coefficient matrix c+1 c (b) For each following cases of c, classify the stability (stable or unstable) and the type (center, node, saddle, or spiral) of the critical point (0,0). Note that if a critical point is a center, it is stablhe. (4) c=흘 (2) c=-2 (1) c=-1 (3) c=-8
Consider an autonomous...
Consider an LTI system whose input x[n] and output y[n] are related by the difference equation y[n – 1] + 3 y[n] + $y[n + 1] = x[n]. Determine the three possible choices for the impulse response that makes this system 1) causal, 2) two-sided and 3) anti-causal. Then for each case, determine if the system is stable or not. Causality Impulse Response Stability Causal Unstable v two-sided Unstable anti-Causal Unstable y In your answers, enter z(n) for a discrete-time...
3. Consider an autonomous system z'=cx +2y where c is a real constant. 2 (a) Calculate the trace T and the determinant A of the coefficient matrix -2 1 (b) For each following cases of c, classify the stability (stable or unstable) and the type (center, node, saddle, or spiral) of the critical point (0,0). Note that if a critical point is a center, it is stable. (1) c--5 (2) c--3 (3) c1 (4) c 6
3. Consider an autonomous...
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