 # 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 is a critical point of the systems.
. Futhermore, it is a center of the corresponding linear system.
b) The systems are almost linear.
c) To prove: hence the critical point for the system is asymptotically stable and the solution of the initial value problem for becomes unbounded as , hence the critical point for the system

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a)
The systems of equations are
Formula used: The points, if any, where $\left[f\left(x\right)=0\right]$ are called critical pointsof the autonomous system $\left[{x}^{prime}=f\left(x\right)\right].$
Proof: The critical points of the system are found by solving the equations
From the equation and from the second equation

Thus, the only critical point is
Now,
The critical points of the system are found by solving the equations