I have a question regarding this differential equation here, this is a picture of the direction field. I am kind of confused is it correct that the only solution that is independent of t is the horizontal line at p = 900? so the line I draw is a solution as well, but it is a solution that depends on both p and t, is it correct?
It says the equilibrium solution is independent of t. But how about the solution I draw here using the red color? (the red line), I thought this solution and basically all other solutions also depend on t in addition to p, is it correct?
And what's so significance if all other solutions converges to the p = 900 if t goes to infinity? and if they all diverge from p = 900? what is the significance of the observation? (divergence vs convergence when t approaches infinity).
Best Answer
We have the differential equation
$$\dfrac{dp}{dt} = \dfrac{1}{2} p - 450$$
An equilibrium (or equilibrium point) of a dynamical system generated by an autonomous system of ordinary differential equations (ODEs) is a solution that does not change with time. Another way of saying this is where the derivative is zero (constant).
To find the equilibrium point(s), we set $p' = 0$, so find $p = 900$.
If we draw a direction field plot of $p'(t)$, it shows a representation of how all solutions would behave. We should see a constant zero slope at the equilibrium point.
This general direction field plot looks like
We can see even more solutions in this variant
Notice that the slope is constants at the equilibrium, $p = 900$, but that above that, it is increasing and below that it is decreasing. Why?
Lets choose two different initial conditions and solve of ODE. The first above the equilibrium and the second below
$$p(0) = 950 \implies p(t) = 50 \left(e^{t/2}+18\right)$$
$$p(0) = 850 \implies p(t) = -50 \left(e^{t/2}-18\right)$$
Lets superimpose those two curves onto the direction field plot (red is the first and green the second)
What do you notice if you take the limit of each of those results as $t \rightarrow \infty$? This is what is so excellent about a direction field (phase portrait) plot, it shows the general behaviors across all solutions.
Now, what do you get if you solve the ODE for $p(0) = 900$?
You can experiment with this direction filed and phase portrait plotter for more examples (just click anywhere for an IC).