Differential Equations and Infinitesimals

Question

Say we have a differential equation such as $ \frac{dy}{dx} =xy$

I was taught you would solve by rewriting the equation as $\frac{dy}{y} = xdx$

Then you integrate both sides $ \int\frac{dy}{y}=\int xdx$

However, what did not make much sense is why you have to make sure that you integrate $y$ with its corresponding infinitesimal: $dy$ and $x$ with its infinitesimal: $dx$.

Even more fundamentally, why do you need to multiply an expression by its infinitesimal to integrate? Why can you not just use any infinitesimal? What makes $dy$ and $dx$ so special in this case?

I understand that it is not technically correct to separate $dy$ and $dx$ so easily but I was wondering if we can get around this by defining

$dy= \lim_{h \to\ 0} f(x+h)-f(x)$ and $dx=lim_{h \to\ 0} h$

or is this not what was implied by Leibniz?

Furthermore, since $y$ is essentially a function of $x$, then is there any such relationship where $dy$ is a function of $dx$ and is this why you have to multiply $y$ by $dy$ and $x$ by $dx$ to integrate so that both sides of the equation are the same?

If the answer has to deal with nonstandard analysis, I would really appreciate it if you explain a little of the basics of the field in your answer so I can get it. Thanks a lot!

No one asked me, but as a student I hated it that my professors told me that $\frac{dy}{dx}$ was not a fraction so don't dare treat it like one, then immediately treated it like one in ODEs. — Randall, Nov 28 '17 at 22:18
Related, possibly helpful https://math.stackexchange.com/questions/2475226/integral-without-differential/2475258#2475258 — Ethan Bolker, Nov 28 '17 at 22:21
I wish I had Mike Spivey's answer here when I was a student: https://math.stackexchange.com/questions/27425/what-am-i-doing-when-i-separate-the-variables-of-a-differential-equation — Randall, Nov 28 '17 at 22:29
Possible duplicate of What am I doing when I separate the variables of a differential equation? — Hans Lundmark, Nov 28 '17 at 22:46
Please, if you are ok, you can accept the answer and set it as solved. Thanks! — user, Jan 10 '18 at 22:17

score 1 · Accepted Answer · answered Nov 28 '17 at 22:21

1

it's only a formal way to obtain the correct result by separation of variables

it's not a rigouros method

take also a look here

Why does the "separation of variables" method for DEs work?

answered Nov 28 '17 at 22:21

user

154,566

score 0 · Answer 2 · answered Nov 28 '17 at 22:24

Sure, you could use any infinitesimal in your integrands, you could integrate y dx if you wanted, but then you'd have to know how to actually integrate y dx. Not knowing the relationship between y and x a priori, this is… difficult to impossible. But you do know how to integrate x dx (into x^2/2 + C) regardless of relationship between x and y, and similarly for 1/y dy, and so on, and that is why this kind of rearrangement is useful.

Differential Equations and Infinitesimals

2 Answers2