Consider the differential equation $x'=x^2-9$
a. find the stability type of each fixed point
To find the fixed points, I set this equal to $0$, right? Would someone mind explaining why I do this? I don't really understand the concept.
So I get the fixed point as being $x=3$ and $x=-3$.
Checking the stability:
$x=2$ results in $-5$ (left)
$x=4$ results in $7$ (right)
$x=-4$ is positive (left)
$x=-2$ is negative (right)
How do I determine their stability exactly?
b. sketch the phase portrait on the line
c. Sketch the graph of $x(t)=ϕ(t;x_0)$ in the $(t,x)$-plane for several representative initial conditions $x_0$.