Question

for wt value of k the following system of equations have unique solution
x-kyequal to 2 ;3x+2y equal to -5​

Answer 1

Answer:

Which are the stable and unstable points of dx/dt=2sqrt (1-x^2)?

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I do not like solving someone's math homework, so I will rather give you bit of explanation.

Your dynamical system is defined by ODE :

dxdt=21−x2−−−−−√,x∈Dx

0. check what could be the domain of your dynamical system.

Due to square root, domain is \big{x \in [-1,+1] \big}

1. find stationary points

0=21−x02−−−−−−√

Which has 2 solutions, -1 and +1. Not it is important to bear in mind that both these points are on the border of the domain.

2. Define ξ as the perturbation from the stationary point. For the perturbation applies:

dξdt=21−(x0+ξ)2−−−−−−−−−−√= 21−x02−2x0ξ−ξ2−−−−−−−−−−−−−−−−√

Now you insert values of stationary points and compute exact value of the derivation.

However, there in this case there is a faster solution. You know that some term with square root will have always non-negative value. Therefore a small positive perturbation from -1 will keep growing. On the other hand, a small negative perturbation from +1 will fade out.

See, in cases like this, you can check stability ans instability of points without actually evaluating whole formula.