Thread: Bernoulli's Insights into Solving Nonlinear Differential Equations

Hello Friends:

Although this opening post delves into differential equations, what I'm really interested in discussing is the insights that great mathematicians come up with to solve problems. So bear with me a bit while I present an example of such insight.

I was recently studying linear differential equations, and the book I'm using asks the reader to show how Bernoulli came up with a technique to transform nonlinear differential equations into linear differential equations making them solvable using the techniques in the book. Bernoulli's technique transforms nonlinear differential equations of the type

y' + P(x)y = Q(x)yⁿ

where n isn't 0 or 1 (if n = 0 or 1 the equation is linear and solvable using the book's methods)

into linear equations

u' + (1 - n)P(x)u = (1 - n)Q(x)

in which u' = y^1-n

After some unfruitful attempts to figure out how Bernoulli came up with this method, I went to a search engine and found that the explanation is that Bernoulli divided the first equation by yⁿ getting

y^-ny' + P(x)y^1-n = Q(x)

Setting u = y^1-n, he then implicitly differentiated both sides getting

u' = (1 - n)y^-ny'

Substituting u'/(1 - n) for y^-ny' and u = y^1-n into y^-ny' + P(x)y^1-n = Q(x) Bernoulli came up with

u'/(1 - n) + P(x)u = Q(x)

Finally, multiplying both sides by (1 - n) we have

u' + (1 - n)P(x)u = (1 - n)Q(x)

which is a linear differential equation that can be used to solve nonlinear differential equations.

I can see several key insights here. The first is the division by yⁿ, the second is setting u = y^1-n, and the third insight is the implicit differentiation resulting in u' = (1 - n)y^-ny'.

What still puzzles me is how Bernoulli came up with these insights. They're not obvious, and I'm not sure if I ever could have discovered them. How do mathematicians come up with such ingenious ideas? Is there something special about Bernoulli's thinking, or can any of us develop insights like this?

I'd appreciate some useful feedback.

Jagella

Hi Jagella,

I think there's really two sides to the issue you bring up. On the one hand, asking the question: "Where do brilliant ideas come from?" in general leads to a collective shrug by most experts. Sometimes the lightening just strikes. On the other hand however, in the overwhelming majority of cases (including the case you mention above) insight comes as a new piece of an already well understood puzzle. Typically large insights appear at first to come out of the blue, completely unprecedented, but then are latter realized to be the next step in a logical whole.

The theory of special relativity is a good example of this. Einstein's paper was groundbreaking and completely unprecedented. It took the world by storm in many ways. However, looking back on it now, many historians of science see many previous works as being "precursors" or "prescient" regarding relativity. It is generally now thought that the world was "ripe" for the theory of relativity to be discovered and if Einstein hadn't written his paper someone in the next decade or so would have stumbled onto it. General Relativity is another matter.

This does not in any way detract from Einstein's brilliance. His insight was phenomenal and, no one shot wonder, he produced many other works including his paper on the photoelectric effect which lead to the foundation of Quantum Mechanics (and somewhat ironically, the one he won his Nobel prize for). However, it does give some insight into how these great leaps forward happen.

Two sources you might find interesting on this are Khun's "The Structure of Scientific Revolutions" and Malcolm Gladwell's "Blink." The first looks at how science advances. This is quite a bit different from mathematics, since mathematics can be proven whereas science advances by theories and the disproving of theories. However, the look at how great ideas form is still applicable. The second source is a look at how our brains work. Although at first glance "Blink" appears to be a book designed to teach you how to use "The Power of Thinking Without Thinking" in actuality it is not. It's really a book about how our brains can sometimes make sudden leaps in understanding, and what precedes these leaps.

In response to your final question, whether anyone can develop these insights, the answer from most sources who have studied this phenomenon seems to be a resounding "Yes." Although at first glance Bernoulli's insight appears quite magnificent. On further inspection we see that it is made of several pieces, none of which is truly ground breaking. He uses substitution of variables, implicit differentiation, and a little algebraic fiddling to reduce the problem to one that can be solved. The insight is quite good, but it comes from a deep knowledge of each of these pieces.

This is essentially the answer found in Gladwell's book. What he discusses is the ability of well trained minds to make these kinds of intuitive leaps when they have a deep understanding and a lot of experience with the pieces that make the insight up. Basically, not everyone can be Einstein and write paper after paper worthy of a Nobel prize, but anyone can prep his or her mind to be able to make these intuitive jumps. By gaining a deep understanding of the underlying mathematics and spending a good amount of time developing one's skills in solving problems, one begins to grow a kind of intuition about such problems. When this happens, the mind is prepped to "Blink" into a new insight about the problem, which in hindsight you might realize was sitting there the whole time (like special relativity) but when it first occurs seems to come out of absolutely nowhere.

A mathematician by trade and training, I leave you with this final thought. I can't count the number of times I have worked for hours upon hours on a problem, considering it's every nuance and attacking it from angle after angle only to give up and take a shower or head to bed. As I've lain in bed just drifting off to sleep, my mind no longer consciously thinking about the problem, the solution will sometimes leap full formed into my brain! I will often jump out of bed and begin scribbling the answer down on the nearest surface. You too can develop such insights. By "priming" your brain with use in mathematics you will eventually develop a mind that is ripe for mathematical insights just as Bernoulli's was.

Originally Posted by MazerRackhem

...askingthe question: "Where do brilliant ideas come from?" ingeneral leads to a collective shrug by most experts.

Yes. It looks like I've had that effect on the experts in this forum. Seriously, I think that much of the way math is done is to learn the insights of the great mathematicians without developing one's own insights.

Sometimes the lightening just strikes.On the other hand however, in the overwhelming majority of cases(including the case you mention above) insight comes as a new pieceof an already well understood puzzle.

I have a book that includes Euclid'sproof that the square-root of two is an irrational number. I noticed that his proof demonstrates his deep knowledge of rational numbers. Evidently such knowledge allowed Euclid to know what is not a rational number. As you say, he already had most of the pieces of the puzzle in place.

The theory of special relativity is agood example of this. Einstein's paper was groundbreaking andcompletely unprecedented.

I understand that Einstein thought visually and independently. Whenever possible, like he I try to extend the concepts and form my own ideas about mathematics. I use drawing software to create diagrams of the material I'm studying. Seeing the concepts in action is very helpful.

Two sources you might find interestingon this are Khun's "The Structure of Scientific Revolutions"and Malcolm Gladwell's "Blink."

I will read both books when I get the time.

Although at first glance Bernoulli'sinsight appears quite magnificent. On further inspection we see thatit is made of several pieces, none of which is truly ground breaking.He uses substitution of variables, implicit differentiation, and alittle algebraic fiddling to reduce the problem to one that can besolved. The insight is quite good, but it comes from a deep knowledgeof each of these pieces.

It appears to me that the steps Bernoulli used fit together in a whole that he may have thought about first. After getting a broad idea of how to solve the problem, he gathered the parts and assembled them into that whole—or at least that's the way I tend to think. If I focus too much on the individual parts I will often lose sight of what I'm trying to do! Nevertheless,you are correct in that understanding the parts is essential to problem solving.

A mathematician by trade and training,I leave you with this final thought. I can't count the number oftimes I have worked for hours upon hours on a problem, consideringit's every nuance and attacking it from angle after angle only togive up and take a shower or head to bed. As I've lain in bed justdrifting off to sleep, my mind no longer consciously thinking aboutthe problem, the solution will sometimes leap full formed into mybrain! I will often jump out of bed and begin scribbling the answerdown on the nearest surface.

That just recently happened to me: Iwas studying the probability of a trained rat entering one of three compartments through openings in a maze. The book I'm using states that one of the probabilities is 2/3. I was stumped until the answer came to me while I was getting ready for bed. It seems that the human brain needs rest so it can more easily carry its work.

You too can develop such insights. By"priming" your brain with use in mathematics you willeventually develop a mind that is ripe for mathematical insights justas Bernoulli's was.

Thanks! I will keep working on that goal.


Jagella