Differential Equation 1consider the autonomous differential

Differential Equation

1.consider the autonomous differential equation:

dx/dt=x^3-9x

a)find all the critical points(equillibrium solutions) of this ODE

b)draw the pulse line(phase diagram) for this ODE

c)determine whether each critical point is stable or unstable.

d) if x(0)= -1. what value will x(t) approach as t goes to infinity.

e)sketch typical solution curves of the given ODE, on the attached graph paper. be sure to includegraphs of all equillibrium solutions, and be sure to label the axes.

Solution

a).Given dx/dt=x³-9x

critical points are given by dx/dt=0

so x³-9x=0

x(x²-9)=0

x(x+3)(x-3)=0

x=0 or x+3=0 or x-3=0

x=0,x=-3,x=3 are the critical points