15.1 Equilibrium solutions
Now that we have our systems of linear equations, let’s understand the dynamics. A first question is to examine the equilibrium solutions, or places where both \(\displaystyle \frac{dx}{dt}=0 \mbox{ and } \frac{dy}{dt}=0\). It might be helpful to imagine what we should expect for an equilibrium solution. Think back to calculus - what types of functions have a zero derivative? (Hopefully constant functions comes to mind!)
If constant functions are a type of equilibrium solution, does that mean any constant function is an equilibrium solution? Let’s try this out with our example:
\[\begin{equation} \begin{split} \frac{dx}{dt} &= 2x \\ \frac{dy}{dt} &= x+y \end{split} \end{equation}\]
If we evaluate our differential equation at \(x=3\) and \(y=5\) (which are both constant solutions), we have the following:
\[\begin{equation} \begin{split} \frac{dx}{dt} &= 2\cdot 3 = 6 \\ \frac{dy}{dt} &= 3+5 = 8 \end{split} \end{equation}\]
So looking at our results the right hand sides of the differential equation do not evaluate to zero. Does that mean our intuition is wrong? Not necessarily - maybe we just didn’t pick the correct solution. Note in the second equation we have \(\displaystyle \frac{dy}{dt}= x+y\) - what if we picked \(x=0\) and \(y=-5\)? That makes \(\displaystyle \frac{dx}{dt}=0\), but \(\displaystyle \frac{dy}{dt}=-5\). Oh no - that is not an equilibrium solution either!
However, a solution that could work is if both \(x=0\) and \(y=0\) (Try to do this on your own.) Here is an amazing fact: it turns out any linear system has the origin as its only equilibrium solution.. I won’t discuss this fact more, but one way to explain this is by examining solutions for linear systems of equations in linear algebra.
You might be wondering why all the fuss with equilibrium solutions - especially the origin (\(x=0\) and \(y=0\))5. So while equilibrium solutions are not terribly interesting question at the moment, the stability of solutions are. In order to understand what I mean by stability let’s re-examine phase planes from Section 6
Another name for the origin equilibrium solution is the trivial equilibrium.↩︎