Are Phase Portraits a big part of Differential Equations?

College Life
1

<p>Okay, so my differential equations class was broken down into four sections.</p>

<ol>
<li>Equilibria, variable separation, bifurcations, integrating factors and phase portraits in one dimension.</li>
<li>Phase portraits of linear systems</li>
<li>Phase portraits of nonlinear systems, and periodic orbits</li>
<li>Laplace transforms</li>
</ol>

<p>Is differential equations really a lot about phase portraits? I look up tests from other institutions and I hardly see anything about phase portraits. I did NOT do very well in this class (math is my strongest subject and the lowest grade I’ve ever gotten is in this class). What do you think differential equations really is about?</p>

2

<p>Was it an ODE class or a PDE class? Since you covered Laplace’s equation I’m guessing it had at least some PDE component.</p>

<p>I had to google “phase portraits.” Never seen them called that before, I just called them vector fields even if that may not be correct (close enough, durnit!). I don’t remember them being much more than an aid to visualizing sets of solutions in the x-y plane or the x-t plane (like where time is the horizontal axis and you can see certain solutions blow up or approach stable values, etc). I think simple examples like that may have featured in some tests or homeworks.</p>

<p>In my physics classes it seems like a lot of the differential equations we solve are either those solvable with simple separation of variables, or are second-order ODEs which are easy to solve with exponentials, or Laplace’s equation stuff using spherical harmonics, and some other stuff I can’t remember now involving boundary conditions. So far (I haven’t hit grad school yet or done stat mech yet) I haven’t had to use the more advanced techniques like second order diffy qs with variable coefficients. There was some other PDE stuff I can’t remember. But I don’t really remember “phase portraits” coming up unless you count orbit problems or bound quantum systems as “phase portrait” problems.</p>

3

<p>Say, given an equation, we would have to draw one of these from scratch:
<a href=“http://www.emeraldinsight.com/content_images/fig/0730110404001.png[/url]”>http://www.emeraldinsight.com/content_images/fig/0730110404001.png&lt;/a&gt;&lt;/p&gt;

4

<p>It wasn’t when I took it, but maybe your class emphasizes intuition. I’m surprised to see nothing about second order differential equations or series solutions. Frankly your class does not seem very good (at best it’s unorthodox, at worst not very rigorous).</p>