To determine: a) The origin [left(0, 0right)] is a critical point of the systems. [frac{dx}{dt}=y + x left(x^{2} + y^{2}right), frac {dx}{dt}= -x + y

nicekikah 2020-11-08 Answered

To determine:
a) The origin [0 0] is a critical point of the systems.
[dxdt=y + x(x2 + y2), dxdt= x + y(x2 + y2)]and[dxdt=y  x(x2 + y2), dxdt= x  y(x2 + y2)]. Futhermore, it is a center of the corresponding linear system.
b) The systems [dxdt=y + x(x2 + y2), dxdt= x + y(x2 + y2)]and[dxdt=y  y + x(x2 + y2), dxdt= x  y(x2 + y2)] are almost linear.
c) To prove: [drdt < 0]and[r 0 as t ], hence the critical point for the system [dxdt=y  x(x2 + y2), dxdt= x  y(x2 + y2)] is asymptotically stable and the solution of the initial value problem for [r with r=r0 at t=0] becomes unbounded as [t12 r20], hence the critical point for the system

You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Expert Answer

falhiblesw
Answered 2020-11-09 Author has 97 answers

a)
The systems of equations are [dxdt=y + x(x2 + y2), dxdt= x + y(x2 + y2) and dxdt=y  x(x2 + y2), dxdt= x  y(x2 + y2)].
Formula used: The points, if any, where [f(x)=0] are called critical pointsof the autonomous system [xprime=f(x)].
Proof: The critical points of the system [dxdt=y + x(x2 + y2), dydt= x + y(x2 + y2)] are found by solving the equations [y + x(x2 + y2)=0 and x + y(x2 + y2)=0].
From the equation [x2 + y2= yx] and from the second equation [x2 + y2=xy],
 yx=xy
 x2 + y2=0
Thus, the only critical point is [0 0].
Now,
The critical points of the system [dxdt=y  x(x2 + y2), dydt= x  y(x2 + y2)] are found by solving the equations

Not exactly what you’re looking for?
Ask My Question

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Relevant Questions

asked 2020-12-02
An exceptional standing jump would raise a person 0.80 m off the ground. To do this, what force must a 66-kg person exert against the ground? Assume the person crouches a distance of 0.20 m prior to jumping, and thus the upward force has this distance to act over before he leaves the ground.
asked 2021-04-25
A stunt man whose mass is 70 kg swings from the end ofa 4.0 m long rope along thearc of a vertical circle. Assuming that he starts from rest whenthe rope is horizontal, find the tensions in the rope that are required to make him follow his circular path at each of the following points.
(a) at the beginning of his motion N
(b) at a height of 1.5 m above the bottom of the circular arc N
(c) at the bottom of the arc N
asked 2021-02-11
A 95.0kg person stands on a scale in an elevator. What is the apparent weight when the elevator is (a) accelerating upward with an acceleration of 1.80 ms2, (b) moving upward at a constant speed and (c) accelerating downward with an acceleration of 1.30 ms2?
asked 2021-01-31
A straight vertical wire carries a current of 1.20 A downward in a region b/t the poles of a large superconducting electromagnet,where the magnetic filed has magnitude B = 0.588 T and is horizontal. What are the magnitude and direction of the magnetic force on a 1.00 cm section of the wire that is in this uniform magnetic field, if the magnetic field direction is a) east? b)south? c) 30 degrees south of west?
According to the solutions, the force stays the same for parts a,b,c:F=IlB=7.06×103 but for part c,why isn't the 30 degree angle used to calculate the force?
asked 2021-11-21
In getting ready to slam-dunk the ball, a basketball player starts from rest and sprints to a speed of 6.0 m/s in 1.5 s. Assuming that the player accelerates uniformly, determine the distance he runs.
asked 2021-03-12

Four point charges form a square with sides of length d, as shown in the figure. In the questions that follow, use the constant k in place of 14πϵ0.

Part A
What is the electric potential Vt at the center of the square?
Make the usual assumption that the potential tends to zero far away from a charge.
Express your answer in terms of q ,d, and appropriate constants.
Vt= 
Part B
What is the contribution U2q to the electric potential energy of the system, due to interactions involving the charge 2q ?
Express your answer in terms of q, d, and appropriate constants.
U2q=
Part C
What is the total electric potential energy Ut of this system of charges? Express your answer in terms of q, d, and appropriate constants.
Ut=
, plz i want just the final answers , thanks

asked 2021-07-04
When attempting jump shots, Myles makes 3 and misses 2 . If he makes 36 jump shots during one practice, hov many jump shots did he miss?

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question