Question

7. Let a system be given by daa) where a is a real number. (a) How should a be chosen to make x = 2 an equilibrium point? (b) How should a be chosen to make 0 an unstable equilibrium point? (c) How should a be chosen to guarantee that any solution starting with o1 will increase without bound? (d) Explain why, regardless of the value of a, no solution to this system can ever decrease [15] without bound.

Answer 1

For an autonomous differential equation,

dx dt = f(x)

the equlibria is at the point where, .

dix

(a) So to make x=2 as an equillibrium point,

f (x) <0

2(2-a)<0 $\Rightarrow$4-2a<0$\Rightarrow$a>2.

So, the value of a should be 2 for x>2 to be an equillibrium point.

(b) For a point x₁ to be an unstable equillibrium point $\frac{\mathrm{dx} }{\mathrm{d} t}>0$ .

so x(x-a)>0 $\Rightarrow$ a<0

So for x=0 to be an equllibrium point a<0.