How to Visualize Differential Equations?

[cowmoo32]

I don't feel that I can truly appreciate a math without being able to visualize it in my head. Generally speaking: calculus flows into areas, trig builds shapes, and linear algebra builds spaces, but I cannot for the life of me look at a diff eq and 'see' it, so to speak. While only proficient in trig and calculus, I'm learning linear algebra and diff eq but I'm finding it harder to grasp diff eq for this very reason.

 

[nlsherrill]

I don't feel that I can truly appreciate a math without being able to visualize it in my head. Generally speaking: calculus flows into areas, trig builds shapes, and linear algebra builds spaces, but I cannot for the life of me look at a diff eq and 'see' it, so to speak. While only proficient in trig and calculus, I'm learning linear algebra and diff eq but I'm finding it harder to grasp diff eq for this very reason.

Trying to visualize the differential equation itself I think is very difficult. I don't know how to do it, unless it represents a physical system where I derived the differential equation from. Visualizing solutions to ODE's at least is much easier. If you can't, then plot it in MATLAB or MAPLE to help you get an idea of what the solution is saying. I suggest looking at a simple mass/spring system, such as this:

http://en.wikipedia.org/wiki/Harmonic_oscillator

 

[rtyabc]

there are things we can solve by mathematical equations but not visualize it...as a proof directly , example we can visuaize (a+b)^2, or (a+b)^3...but can you visualize (a+b)^4 that easily but we can get a solution that's the beauty of mathematical expressions

 

[robphy]

Phase plots might help.
For example, from google,
http://tutorial.math.lamar.edu/Classes/DE/PhasePlane.aspx

From further down on a google search,
http://www.dump.com/2011/05/18/a-cool-way-to-visualize-differential-equations-video/

 

[dalcde]

I don't like to visualize them. Instead, I try to understand them. For example, with the equation
$$\frac{\mathrm{d}y}{\mathrm{d}x}=x,$$
I think that how fast y changes is proportional to x. If I have
$$\frac{\mathrm{d}y}{\mathrm{d}x}=x+x^2,$$
I think that there are 2 factors that contribute to the change of y - x and $x^2$.

 

[Eynstone]

Visualisation in the phase space or as a vector field might be helpful. However, visualisation in dimensions higher than 3 is difficult & can be misleading.
Have you heard this one ? - A biologist is asked what his mental picture of a dog is. He says that the picture is as follows:

DOG

I think this is a little more than a mere joke.

 

[Jaynte]

Something all are familiar with are speed and acceleration which are
simple examples of diff. equations.

For example:

$$v=\frac{dx}{dt}$$
$$a=\frac{dv}{dt}=\frac{d²x}{dx²}$$

where the velocity v is how long distance you got after a certain time. Or in other words, difference in distance over difference in time.

And same for acceleration which describes how the speed has changed after a certain time.

Don't know how that helps visualize it but you can feel acceleration :)

 

[pmsrw3]

DEs come in a variety of sizes and shapes, so it's hard to give a general answer to your question. But there are helpful visualization methods for some common classes of DEs. Strogatz describes some of them in Nonlinear Dynamics and Chaos.

 

[nickalh]

I. In Diff. Eq. there are a variety of methods for a variety of equations.

Something to consider:
Create a summary table showing
A. what form the original equation looks like.
B. steps in the solution method.
C. Other notes or cautions to be aware of, possible comparisons & contrasts.

If you lay these out in a clear pattern, it can help one visualize the course, & understand the varieties of methods to apply & when to apply them.


II. Unfortunately, when starting a Diff. Eq. the visualization options tend to be limited to computer methods. I'm more familiar with the terms
"vector field" or "slope field"
See http://www.sosmath.com/diffeq/slope/slope1.html"
Any college level mathematical software will have and thankfully, wolframAlpha.com may be able to graph vector fields.
See
http://reference.wolfram.com/mathematica/ref/VectorPlot.html"


III. When solved it can be very instructive to plot a "solution family" or multiple solution curves. Using a different color for each solution can be useful.
http://www.wolframalpha.com/input/?i=2y'+++xy+=+sin+x" Notice the "sample solution family", where x:0->2.

Bye.