Expert: Daniel Mazur - 7/7/2009

Question
Hi Daniel,

I had some conceptual problems with differential equations. Though I can solve them and am familiar with all the methods of solving linear equations of degree 1, but had a few points that I'm feeling a little uncomfortable about.

Problem 1
----------

If an equation says dy/dx=4x^2, I understand it as the value of y changes with value of x and this change in value of y varies from one x to the other.
So at a particular x, the change in y per change in x is 4x^2.

But there are also equations which can be solved through variable seperable method, i.e. they are explicitly or implicitly of the form,
dy/dx=f(x)/f(y) or dy/dx=f(y)/f(x)
And we can now get the terms of the same variable on one side and integrate both sides.

What I fail to understand over here is that how can the change in y with change in x depend on both x and y??

Either dy/dx=f(x) or dy/dx=f(y)   [both tell us the
                                  same thing for a
                                  particular function]

But I can not understand, that for a particular function, how can dy/dx depend on both 'y' and 'x' ??? Or in other words, how is dy/dx stated in terms of both x and y?
dy/dx varies from point to point. So we can state it in terms of either different y's or different x's but how can we or why do we state it in terms of both x and y in the above sort of differential equations??


Problem 2:
-----------
While solving linear diff. equations of the type

dy/dx + Py = Q    -------------- (1)

we find the integrating factor, multiply it on both sides and then integrate.
The procedure of finding the Integrating factor(I.F.) or the reasoning behind it is as follows (given in a book):

We multiply both sides of the above equation with a func. g(x). We chose g(x) in such a way so that the R.H.S. of the above linear diff. equation, becomes d{y.g(x)}/dx

So we equate the L.H.S. i.e. { g(x)dy/dx + g(x)Py } to  
d{y.g(x)}/dx to find this func. g(x) which will make the R.H.S. d{y.g(x)}/dx.


This func. is the I.F. and is given by

g(x) = e^integ.{P.dx} = I.F.
On now multiplying the eq. (1) with this factor, the L.H.S. becomes the derivative of some func.of x and y and then we can integrate and solve the equation.

I just didn't understand why we need to chose a g(x) which makes the R.H.S. "d{y.g(x)}/dx", and then how by multiplying this function throughout on the eq. (1), the L.H.S. of the equation becomes a derivative of some func. of x and y. I've verified that this does happen but don't understand how??

I want to know how can we understand this method of solving equations, logically. Is it just that mathematicians sat down to solve this type of an equation and one just found this method purely by hit and trial. I don't think that would have been the case. There must be a reason as to the steps. That's what I'd like to understand.

And especially how do we know that if we find a func. g(x) that will make the R.H.S. "d{y.g(x)}/dx" on multiplying, we will get some func. of x and y on the L.H.S. later when we multiply (1) with this g(x)...????

Thanks,
Shikhin

Answer
Shikhin,

Ad problem 1:
The reason many equations have inseparable variables is the same as why implicit equations exist: the "elementary functions" we are most comfortable with (powers, exponentials, polynomials,...) only form a small subset of the set of "all functions". A differential equation with inseparable variables has a solution that cannot be expressed as a finite function of elementary functions. Mind you, some equations may be separable by a trick that you and I just don't know...

If you look at it the opposite way, it is easy to construct a DE, where dy/dx depends both on x and y. For example, y'=4x^2 has a solution y=(4/3)x^3. If you wish, you can with this knowledge write the DE as y'=3y/x. This is a way to manufacture a problem, but in fact Nature and our varied human perspectives give us plenty of opportunities to describe a problem by a DE that is like that. Example: Describing velocity (y') in terms of time (x) as the ONLY free variable is useful, if your primary interest is observing energy changes in your system. But if you want to see if an asteroid collides with the Earth and what effects the impact will have, you will need to work with time (x) AND coordinates (y) to construct a complete equation. Frequently, what is useful is not necessary simple...

Ad problem 2:
The answer is, to all I remember, that tricks like this one ARE frequently discovered by hit and miss, trial and error or simply by chance. I don't mean "chance" like just dreaming about it or putting the terms down as in lottery, I mean the a mathematician would, in his or her line of work, "play" with a more general problem and a trick like this suddenly comes out as a special case. Or on other times, a mathematician would study the properties of terms like d{y.g(x)}/dx and at some point they find the significant property that it leads to an integration factor of a certain type of linear DE.

It is usually impossible to tell, how a person discovered a law of nature or a trick to solve an equation. Scientific articles will tell you, for example, how a certain novel material as been obtained by quench (rapid cool down) of a melt. They will never tell you, though, that the only bit of the material that really had interesting properties was actually a droplet of melt accidentally spilled on the laboratory floor. This actually happened in a group I once worked in. If there was a logical way, how to derive discoveries, you would not need humans to do research, machines could do it. Many times we would like to know what led to one discovery or another, but it just isn't possible. I am sure you will hear stories of accidents and lucky chances, but there are only a few around. Too few, sorry.

Cheers,
Daniel