What does constant terms for complex roots represent when drawing a phase portrait?

Question

For example if I have ODE where $x$ is a matrix of $x_1$ and $x_2$ $$x_1 ' = x_1 - 5 x_2$$ $$x_2 ' = x_1 - 3 x_2$$

and I found the solutions which are $$x_1 = c_1 e^{-t} (2 \cos(t)-\sin(t)) - c_2 e^{-t} (2 \sin(t)+\cos(t))$$ $$x_2 = c_1 e^{-t} \cos(t ) - c_2 e^{-t} \sin(t)$$ $c_1,c_2$ are constants

Now I know that due to $e^{-t}$, it will have decaying solution as t goes to infinity and will be unstable spiral that is moving in clockwise direction. However, my question is about $c_1$, $c_2$, what do these $c_1$, $c_2$ represent in the phase portrait?

Answer 1

Those constants provide information regarding your position at time $t = 0$, and further they are entirely dependent on that initial condition. Notice that when you plug in $t = 0$ into your equations for $x_1$ and $x_2$, you are at the point $(c_1, c_2)$.

Answer 2

I changed variable names slightly.

We are given:

$$x' = x - 5\cdot y \\y' = x - 3\cdot y$$

The solutions with no initial conditions are given by (I chose different eigenvectors, so yours are probably okay too):

$$x(t)=c_1 e^{-t} (2 \sin (t)+\cos (t))-5 c_2 e^{-t} \sin (t)\\y(t)=c_1 e^{-t} \sin (t)+c_2 e^{-t} (\cos (t)-2 \sin (t))$$

If we draw a direction (vector) field plot of this system we get:

If we decide to choose a bunch of initial conditions and lay them over the direction field plot, we get the following phase portrait:

Lets choose a pair of initial conditions $x(0) = -1, y(0) = -1$, and our solution becomes:

$$x(t)=-e^{-t} (\cos (t)-3 \sin (t)) \\ y(t)=e^{-t} (\sin (t)- \cos (t))$$

If we were to parametrically plot $x(t)$ versus $y(t)$ (note that at time equal zero, we have $(x,y) = (-1,-1)$, our initial condition and as $t$ increases we have $x(t)$ and $y(t)$ approaching zero in the plot), we would have (compare to phase portrait above):

If we now redraw the Phase Portrait with this initial point (see red curve, and compare it to the parametric plot above), we have:

All of the other solutions on the phase portrait are just a bunch of different initial conditions to fill in the phase portrait. The green one is another sample initial condition for you to work with and see if you can parametrically plot it with $x(0) = 0, y(0) = 3$ (this is your question regarding $c_1 = 0, c_2 \ne 0$).

So, by varying the values of the initial conditions, you are just going to be starting at different places on the phase portrait curves.