Differential Equations: Seeking Advice to Make It Interesting

[DR13]

Hey all,

So I need some advice on my differential equations course. Last semester I took Calc 3 and loved it. I loved that I could visualize everything. It made math seem so much more concrete. On the other hand, there is differential equations. Now I know that there are an immense amount of applications for differential equations (population models, economics, upper level physics, etc). However, we just kinda do them without going too far into the applications. It gets tedious really quickly.The math isn't even that difficult; it's just boring! Does anyone have some advice on how I could make diff eq more interesting?

Thanks,
DR13

 

[snipez90]

Vladimir Arnold's ODE text presents the theory from a geometric perspective, but I would avoid this unless you are comfortable with linear algebra, basic real analysis and basic topology. Birkhoff and Rota's text provides a lot of intuition for ODE theory, but again unless you're familiar with basic analysis, it's hard to appreciate the text.

As for applications, it really depends on what you're interested in. Obviously, if you look through even a basic treatment of calc-based physics, say via Halliday and Resnick's text, you're going to encounter ODEs a lot. For economics, one area that uses differential equations pretty heavily is economic growth. Here it's probably best to start with the Solow growth model, which is probably the most well-known http://en.wikipedia.org/wiki/Exogenous_growth_model" . If you're interested and want access to Solow's original paper, feel free to PM me (though it should be pretty accessible). Unfortunately, I don't know much about the applications of DEs to mathematical biology (basic population models are pretty standard but boring examples).

Since you've taken calc 3, you can also start looking at basic linear PDE theory. I think Walter Strauss's text doesn't require too much analysis, so that may be a starting point.

 

[Delong]

Something that helps me is to make it into a competition. Try to do extra diff EQ problems on your own at home. Pick an especially hard one that goes beyond what you learned in class and challenge yourself to solve it. Wrestle with it and when you finally solve it it's one of the best feelings in the world. Then go show off to your professor and classmates how smart you are.

Applications may make you more motivated to study math but it probably won't make it more enjoyable. Plus I'm in Diff EQ right now and I think I should take my own advice < )

 

[AlephZero]

Almost every "real world" application of DEs has to be solved numerically. The equations that can be solved analytically usually need a huge amount of effort to construct series solutions to match the boundary conditons, etc, which is rather pointless since a computer can get numerical answers more reliably than people can do the math.

The real aim of a DE math course should be teaching you general theorems about the existence and uniqueness of solutions, etc, not lots of tricks for solving particular types of equations. Maybe it doesn't seem interesting because you haven't got to the mathematical "red meat" of the subject yet.

 

[General_Sax]

Yeah, I'm in the same boat. Partial fraction expansions are too boring. However, It's kind of fun to think of the 'why' instead of just memorizing procedure.

 

[MaxManus]

Almost every "real world" application of DEs has to be solved numerically. The equations that can be solved analytically usually need a huge amount of effort to construct series solutions to match the boundary conditons, etc, which is rather pointless since a computer can get numerical answers more reliably than people can do the math.

The real aim of a DE math course should be teaching you general theorems about the existence and uniqueness of solutions, etc, not lots of tricks for solving particular types of equations. Maybe it doesn't seem interesting because you haven't got to the mathematical "red meat" of the subject yet.

True for the real world, but only numerical solutions may not be the best way to learn physics/economics/biology/...?

 

[Leveret]

Then go show off to your professor and classmates how smart you are.

Maybe the culture where you are is different, but to me, that seems like a good way to alienate all your classmates. Don't be "that guy," no one likes a show-off (professors included, if it's too blatant).

Back to the original poster, are you currently in a physics or engineering course? Even if you've only been through an intro level course, you might find it interesting to work through some mechanics problems using differential equations instead of the simplified calculus found in a lot of intro physics textbooks.

 

[chiro]

Hey all,

So I need some advice on my differential equations course. Last semester I took Calc 3 and loved it. I loved that I could visualize everything. It made math seem so much more concrete. On the other hand, there is differential equations. Now I know that there are an immense amount of applications for differential equations (population models, economics, upper level physics, etc). However, we just kinda do them without going too far into the applications. It gets tedious really quickly.The math isn't even that difficult; it's just boring! Does anyone have some advice on how I could make diff eq more interesting?

Thanks,
DR13

It's just another way of describing some system, except that the system is generalized to include differentials of higher order starting from simple equations like exponential ODE's to things like wave-functions and chaotic systems that are just as complex to represent as their behavior.

I agree with you completely to how boring the subject is. As has been stated above, most models have to be solved numerically and learning a bunch of tricks to solve the more "trivial" models seems like a waste of time.

One thing that I should point out is that a lot of systems in nature use information based on what the state of the system is and that should be the motivation behind DE's. Also given this point, a system that has the potential to change either up or down at any time is an infinite polynomial (transcendental function) since if it can have infinite local maximums or minimums means that it must be a function of that nature.

In saying the above, we have found especially only recently that we have found a lot of different transcendental functions outside of the standard exponential/logarithmic (and also hyperbolic functions) and also outside of linear wave functions and their inverses (ie sin/cos/tan etc).

Personally (and this is just my opinion), but I see the future of mathematics describing new classes of infinite series where libraries of functions are built up to describe arbitrary systems that are more complex than ones currently studied with regard to Fourier transforms.

The kinds of functions I'm talking about with respect to Fourier transforms are things like periodic discontinuous functions (triangle,sawtooth,signum etc) and the variety of signal types that we currently analyze.

Understanding infinite series in my mind is key to developing the future of mathematics.

 

[Delong]

Maybe the culture where you are is different, but to me, that seems like a good way to alienate all your classmates. Don't be "that guy," no one likes a show-off (professors included, if it's too blatant).


well yeah I don't suggest doing it in a way that is annoying or alienating. I'm thinking of in the context of like a class where everyone cares about math. Like a math club or a very enthusiastic math class. Maybe showing off is not a good way to put it but it would be like sharing a discovery. Oh well that was the environment I mostly grew up in while learning math. Math competitions and the like.

 

[DR13]

Almost every "real world" application of DEs has to be solved numerically. The equations that can be solved analytically usually need a huge amount of effort to construct series solutions to match the boundary conditons, etc, which is rather pointless since a computer can get numerical answers more reliably than people can do the math.

The real aim of a DE math course should be teaching you general theorems about the existence and uniqueness of solutions, etc, not lots of tricks for solving particular types of equations. Maybe it doesn't seem interesting because you haven't got to the mathematical "red meat" of the subject yet.

Hopefully this is it. I have to take another Diff Eq course my Junior year. It is PDE based on advanced math for engineers so that should be more interesting.