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Introduction to Differential Equations (2008)

tutorial.math.lamar.edu

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Re: Introduction to Differential Equations (2008)

#71

Post by jshowa1 »

craig_asp wrote: post #6
acjohnson55 wrote: post #4

DiffEq seemed like black magic to me when I took it as a freshman in college. They basically just taught us the recipe bag for solving equations in different shapes, but very little insight. When I started a gamified music discovery company, I actually ended up using DiffEq to define a scoring algorithm that would produce continuously varying point values based on time series input data. I had to relearn how to do it…

Yeah, it's relatively dry material which is hard to grasp without the context of why we need it and how it can be applied in the real world. While the intro is good, it still has the same problem - just shows (in a good way) "some math".

You need to learn the math before applying it. You don't apply things that you don't understand.

Re: Introduction to Differential Equations (2008)

#72

Post by jshowa1 »

freetime2 wrote: post #19
acjohnson55 wrote: post #4

DiffEq seemed like black magic to me when I took it as a freshman in college. They basically just taught us the recipe bag for solving equations in different shapes, but very little insight. When I started a gamified music discovery company, I actually ended up using DiffEq to define a scoring algorithm that would produce continuously varying point values based on time series input data. I had to relearn how to do it…

"Ten lessons I wish I had learned before I started teaching differential equations" is relevant here. I feel that DiffEq was the most useless undergraduate course that I took for my comp sci degree. They really didn't spend enough time going into the fundamental concepts so that I am not even sure I could recognize a differential equation if it were staring me in the face at this point... much less any of the tricks…

"Such a teacher will be successful if at the end of the course every one of his or her students feels they have taken “a good course,” even though they may not quite be able to pin down anything specific they have learned in the course."

Basically the equivalent of, if the course makes you feel good, then the course is good. Not something that I'd advocate.

By the way, a differential equation is simply an equation with a derivative in it. If you can't recognize that, then you didn't go far enough in math.

Re: Introduction to Differential Equations (2008)

#73

Post by Fomite »

cgriswald wrote: post #60
craig_asp wrote: post #6

Earlier quoted context omitted.

Yeah, it's relatively dry material which is hard to grasp without the context of why we need it and how it can be applied in the real world. While the intro is good, it still has the same problem - just shows (in a good way) "some math".

We had a great professor and this was one of the most enjoyable classes I've ever taken. One particular assignment was a group paper where we were supposed to essentially explain and use the SIR model. We extended the model to an SIRZ model and effectively argued that zombie apocalypses in fiction are essentially impossible unless they include some supernatural elements. Under a wide range of assumptions ( e.g. , zom…

I use zombies in my epidemic modeling class - one of my favorite results is, with a semi-complex model, you can replicate the script of most zombie movies mathematically (lots of people die, the survivors take shelter somewhere and are safe for awhile, attrition starts to take hold, things collapse and then you're left with a small surviving fraction of protagonists at the end).

Re: Introduction to Differential Equations (2008)

#74

Post by tanderson92 »

enriquto wrote: post #38
RivieraKid wrote: post #21

Can anyone compare this vs other resources to learn differential equations? I want to learn math roughly to the level of an undergrad engineering student, so I've looked at some Advanced Engineering Mathematics books (one by Zill, another by Kreyszig), both have mostly good reviews, and to be honest, Paul's Notes seem almost a level above in clarity and understandability. For example, compare the explanation of integ…

> compare the explanation of integrating factor and exact equations those are fringe subjects, completely irrelevant for the modern usage of differential equations. They are useful only in computer algebra when you want to implement differential galois theory. In practice you want to understand the overall behavior of your system (qualitative theory) or compute particular solutions numerically (using numerical method…

Integrating factors are important motivations in the design of some numerical algorithms. Some keywords: matrix exponentials, semigroup theory, exponential integrators.

Re: Introduction to Differential Equations (2008)

#75

Post by Fomite »

_jal wrote: post #49
Cobord wrote: post #40

Earlier quoted context omitted.

The algorithms you would learn for graphs etc are also just math and equally abstract. What makes the difference in how real you treat the two concepts that are both abstract but generally applicable?

Not the OP, but I've had good and bad math teachers. The bad ones tend to teach rote steps, "do this, do this, do this, done," without any attempt to explain why things work the way they do, without drawing parallels to already-learned things, without trying to teach any _why_. Then you hear students asking, "when am I ever going to use this?" My good teachers, on the other hand, always tied what we were doing into a…

This has a huge impact for me.

I didn't get calculus. It was a disaster for me.

Now I'm a mathematical epidemiologist. Why? Because someone introduced me to the grander scheme of things.

Re: Introduction to Differential Equations (2008)

#76

Post by Sharlin »

acjohnson55 wrote: post #4

DiffEq seemed like black magic to me when I took it as a freshman in college. They basically just taught us the recipe bag for solving equations in different shapes, but very little insight. When I started a gamified music discovery company, I actually ended up using DiffEq to define a scoring algorithm that would produce continuously varying point values based on time series input data. I had to relearn how to do it…

> They basically just taught us the recipe bag for solving equations in different shapes, but very little insight.

This is why I just dropped off my DiffEq class. It was optional anyway, but when I go to a university level math class I expect insight, not rote memorization.

Re: Introduction to Differential Equations (2008)

#77

Post by acjohnson55 »

Cobord wrote: post #40
acjohnson55 wrote: post #4

DiffEq seemed like black magic to me when I took it as a freshman in college. They basically just taught us the recipe bag for solving equations in different shapes, but very little insight. When I started a gamified music discovery company, I actually ended up using DiffEq to define a scoring algorithm that would produce continuously varying point values based on time series input data. I had to relearn how to do it…

The algorithms you would learn for graphs etc are also just math and equally abstract. What makes the difference in how real you treat the two concepts that are both abstract but generally applicable?

Depends on how they're taught, but to be honest, at a certain point in algorithms class, the subject matter also became tedious and abstract beyond recognition.

Re: Introduction to Differential Equations (2008)

#78

Post by acjohnson55 »

jshowa1 wrote: post #71
craig_asp wrote: post #6

Earlier quoted context omitted.

Yeah, it's relatively dry material which is hard to grasp without the context of why we need it and how it can be applied in the real world. While the intro is good, it still has the same problem - just shows (in a good way) "some math".

You need to learn the math before applying it. You don't apply things that you don't understand.

There are certainly people who can learn that way, but it's not effective for most people. Most people have a limit to the amount of abstraction they can operate under before they need a tangible connection. Once that connection is made, most people can continue on to higher levels of abstraction.

Re: Introduction to Differential Equations (2008)

#79

Post by amelius »

jshowa1 wrote: post #69
amelius wrote: post #32

Earlier quoted context omitted.

Also, what they usually don't tell you is that the recipe bag only works for toy problems. For real applications you most often need numerical approximations.

If you consider analyzing an RL/RC circuit as a "toy problem" then I guess you're right.

Linear circuits are mostly analyzed using the Laplace transform, i.e. in the s-domain, where the differential equations are abstracted away. In the time-domain, simulators are still used most often. But yes a really simple circuit like RC/RL is usually done on the back on the envelope, but then you're talking really simple.

The problem with the analytical approach to differential equations is that it doesn't scale well, and you don't know beforehand whether the approach will work, so you might as well use the numerical approach from the start.

Re: Introduction to Differential Equations (2008)

#80

Post by jshowa1 »

acjohnson55 wrote: post #78
jshowa1 wrote: post #71

Earlier quoted context omitted.

You need to learn the math before applying it. You don't apply things that you don't understand.

There are certainly people who can learn that way, but it's not effective for most people. Most people have a limit to the amount of abstraction they can operate under before they need a tangible connection. Once that connection is made, most people can continue on to higher levels of abstraction.

The problem is math is entirely abstract. So you must learn the abstraction before applying it. Otherwise, you learn how to add 2 + 3, but you don't learn how to add n + m.
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