Why differential equations? When to use them?

[Pulik]

Hello everyone,

I have some questions that have been bothering me for years now.

Why do we use differential equations? When would I normally think about using one? What would make me use one?

I know it means the slope of a function and I know the derivation/integration rules + the solution of some ODE's, but I'm missing something. I don't have a feeling for when to use it, and no professor managed to explain it nice. I would appreciate very much a good pedagogical answer.

Paul

 

[S.G. Janssens]

Last week someone asked an almost identical (but valid) question, see https://www.physicsforums.com/threads/what-are-differential-equations-good-for.852064/. In post #2 of that topic, @Runei sketched a classical application. Post #3 refers to an interesting contemporary example modeling using partial differential equations.

I recommend you have a look at https://www.amazon.com/gp/product/0813349109/?tag=pfamazon01-20 by Strogatz. Normally I would always prefer to recommend https://www.amazon.com/dp/0123820103/?tag=pfamazon01-20 by Hirsch, Smale and Devaney, but if you are still unsure about the motivation of differential equations (and, I infer from this, are just beginning to explore this field), then perhaps Strogatz' book is preferable. Just bear in mind that there is a lot more to the subject than he shows and his text is far from mathematically rigorous.

 

[HallsofIvy]

Rather than saying the derivative "means the slope of a function" it would be better to think "the derivative is the rate of change of a function". "Differential equations" occur whenever we have information about how a quantity is changing rather than the quantity itself. For example, if we know how fast a population is increasing but want to find the population itself, we need to solve a differential equation. The fundamental theorem of mechanics is that "force equals mass times acceleration". "Acceleration" is the derivative of the velocity which is itself the derivative of the position function so knowing the force applied to a given mass we have a second order differential equation to solve for its motion.

 

[jackmell]

Why do we use differential equations? When would I normally think about using one? What would make me use one?
Paul

. . . oh goodness. Once upon a time, a long time ago I would look outside of my window at the world about me and wonder why. Then I started studying non-linear differential equations. I no longer wonder why about a lot of things. Many people go through their lives puzzled about why are things the way they are in this world. This causes a lot of stress and unhappiness in their lives. It is a surprise that mathematics can provide comfort and solace to a troubled heart. But if you study differential equations long enough, a grand (non-linear) understanding of the way the world works emerges. And although you may not like some of the things you see, at least you understand why they are the way that they are. This, in my humble opinion, gives great comfort to one's life. Here's a not-so-common example: something bad happens to a family member. Those around him say, "I had no idea he was capable of doing this!". But the sad fact is that he probably "snapped". Well, the concept of "snapping" is a universal phenomenon in Nature, a critical-point is reached in the dynamics of the system, causing a qualitative change. Such critical points are common in non-linear differential equations and the human brain is massively non-linear. Why is there so much diversity in life? Are you familiar with the Lorenz attractor (the owl-eye icon of Chaos Theory) ? Well, that's a system of three non-linear differential equations and the trajectory in phase space of that system NEVER crosses. In some ways it has infinite diversity! And that's just three equations! And Stephen Gould proclaimed, "life is massively non-linear." Why are we so different from apes yet share 98% of their DNA? Are you familiar with the "butterfly" effect? The smallest of changes in a non-linear system can have the most profound consequences. Theoretically (but perhaps not practically) we could be 99.99% similar yet be as different. How does one create a Universe? There is a branch of mathematics called Catastrophe Theory focusing on the critical-point trajectories I described above. Consider the cubic non-linear differential equation:
$$y^3+ay^2+b=0$$

This equation exhibits the cusp-catastrophe. Like pushing a vase across the top of a table. Push it along and nothing much happens, it just moves a little. But get to the very edge, just balancing and the slightest small shove, causes a qualitative change in the system as the vase trajects towards the floor and smashes in a big bang! :)

 

[S.G. Janssens]

Consider the cubic non-linear differential equation:
$$y^3+ay^2+b=0$$

This equation exhibits the cusp-catastrophe.

You mean: the equation defining the stationary points of the two-parameter ODE $\dot{y} = y^3+ay^2+b$?

 

[jackmell]

You mean: the equation defining the stationary points of the two-parameter ODE $\dot{y} = y^3+ay^2+b$?

Ok. Thank you kindly for that. Yes. I'm having problems with it. Also, I think perhaps the one I'm most familiar with is:
$$\frac{dy}{dt}=-y^3+by+c$$

I was wondering then since I'm having problems with it, would someone be so kind to generate the catastrophe surface, identifying the cusp-catastrophe and then using this model, propose a possible mechanism causing the Cambrian Explosion 600 million years ago when life forms on Earth changed, in what is believed to have been a relatively short geological period, from bacteria and single-celled organisms, to most of the multicellular organism forms we see today. Would be a nice example of what differential equations are used for. :)

 

[S.G. Janssens]

Ok. Thank you kindly for that. Yes. I'm having problems with it. Also, I think perhaps the one I'm most familiar with is:
$$\frac{dy}{dt}=-y^3+by+c$$

I was wondering then since I'm having problems with it, would someone be so kind to generate the catastrophe surface, identifying the cusp-catastrophe and then using this model, propose a possible mechanism causing the Cambrian Explosion 600 million years ago when life forms on Earth changed, in what is believed to have been a relatively short geological period, from bacteria and single-celled organisms, to most of the multicellular organism forms we see today. Would be a nice example of what differential equations are used for. :)

I can wholeheartedly recommend http://www.scholarpedia.org/article/Cusp_bifurcation which includes the surface you are after.

Catastrophe theory is not without controversy. I recommend looking in section 3.6 of Strogatz' book for not only a more elementary discussion, but (at the end of the section) also some pointers to the polemical literature.

 

[jackmell]

I can wholeheartedly recommend http://www.scholarpedia.org/article/Cusp_bifurcation which includes the surface you are after.

Yes. Thanks for that. It's certainly the surface but it's the "how" part, precisely, "exactly how" part. That is, given $$y'=-y^3-by+c$$, generate the catastrophe surface preferably in Mathematica in an elegantly-pleasing format (a nice-looking graph that clearly shows the cusp). Can we plot it in a single-line command in Mathematica? Can't remember if it can be done using just one plot command. Just suggesting it if someone is interested in the matter.

 

[S.G. Janssens]

Yes. Thanks for that. It's certainly the surface but it's the "how" part, precisely, "exactly how" part. That is, given $$y'=-y^3-by+c$$, generate the catastrophe surface preferably in Mathematica in an elegantly-pleasing format (a nice-looking graph that clearly shows the cusp). Can we plot it in a single-line command in Mathematica? Can't remember if it can be done using just one plot command. Just suggesting it if someone is interested in the matter.

I'm a Maple user, but I think in Mathematica you could use http://reference.wolfram.com/language/ref/ContourPlot3D.html to plot the equibrium surface with a one-line command. Afterwards you could probably also add things like projections onto the $bc$-plane to get the the cusp curves in the parameter space, as in Figure 1 in the Scholarpedia reference. (However, I know that that figure was in fact drawn using Xfig .)

 

[jackmell]

I'm a Maple user, but I think in Mathematica you could use http://reference.wolfram.com/language/ref/ContourPlot3D.html to plot the equibrium surface with a one-line command. Afterwards you could probably also add things like projections onto the $bc$-plane to get the the cusp curves in the parameter space, as in Figure 1 in the Scholarpedia reference. (However, I know that that figure was in fact drawn using Xfig .)

But I want someone else to do it. Well, I've done it plenty of times, even the swallow-tail for that matter but I digress. No, I want someone to be taken by this discourse and change history forever. :) Butterfly effect right? :)

 

[S.G. Janssens]

But I want someone else to do it. Well, I've done it plenty of times, even the swallow-tail for that matter but I digress. No, I want someone to be taken by this discourse and change history forever. :)

Ah sorry, I thought you yourself were wondering about it.

 

[jackmell]

Ah sorry, I thought you yourself were wondering about it.

And sorry if I misled you. No, what I think would be nice is for someone to take interest in this and generate the equilibrium surface and then add something: draw little bacteria and single-celled organisms on the top surface (the pre-Cambrian) and then trajecting through the catastrophe point right below them, on the bottom surface, flora and fauna of the Cambrian. And if someone has a better possible mechanism explaining the event, I'd like to hear it. Just a not-so-usual example of what differential equations could be used for in the spirit of the this thread title. :)