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)y^{n }

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 y^{n}getting

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

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

u' = (1 - n)y^{-n}y'

Substituting u'/(1 - n) for y^{-n}y' and u = y^{1-n}into y^{-n}y' + 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 y^{n}, the second is setting u = y^{1-n}, and the third insight is the implicit differentiation resulting in u' = (1 - n)y^{-n}y'.

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