Consider the following one-dimensional system $$\frac{dx}{dt}=x^2-1.$$ Then, using the phase portrait, sketch the graph of the solution $x(t)$ for various initial conditions.
I just wanted to confirm whether this means to solve for $x$ using separation of variables and then plotting the curve and varying the $c$ values that come from it? I'm not sure how to show you the Desmos plots otherwise I would've but is that the case for questions like this?
But then how does this use the phase portrait? I have got two fixed points for this system, where $1$ is unstable and $-1$ is stable.
Thanks in advance!
You do NOT need to solve the system to sketch the phase space portrait. At each point int the $\mathbb{R}^2$, say $(x,t)$, you draw and arrow in the direction $(1, \frac{dx}{dt}) = (1,x^2-1)$. So for example, the fixed points $x=\pm 1$ will give you straight lines in the phase space.