Sketch the graph of the solution with various initial conditions.

Question

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!

But how does that use the phase portrait? That's the thing that is kinda throwing me off — thesmallprint, Jun 05 '20 at 16:05
So the curves that I get from varying $x$ and $t$ would be kind of thing I am looking for? Thanks a bunch if so. — thesmallprint, Jun 05 '20 at 16:31

score 1 · Answer 1 · answered Jun 05 '20 at 16:08

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.

Sketch the graph of the solution with various initial conditions.

1 Answers1