B Elementary question on integrating an equation

[jonjacson]

It is a very simple question.

If we have an expression like this one:

x + y = 2

And we have to differenciate it, there is an algorithm that tells us how to do it. We have to find the relationship between the differentials of the given functions. To find them we have to substract the infinitesimal increment of the function minus the function itself:

(x+dx) + (y+dy) - (x + y) = (2+0) -(2), Since 2 is a constant its differential is 0, we found:

dx + dy = 0

It is clear how the differentiation operation acts on every quantity on the equation.

But now we have this expresion:

dx + dy = 1

And we need to integrate it, I understand that we have to find the relation between y and x, whose differentials will make this equation correct. But it is not clear to me how to act on the number 1.

∫dx +∫dy = ∫1

On the left we get x+y + constant, but What happens on the right? Should we integrate on the variable x? Or should it be done on the variable y? How should we proceed?

Thanks!

 

[Delta2]

This equation is improper, it doesn't have a solution. With simple words it is impossible to add two infinitesimal quantities dx and dy and get a constant like 1.

manipulating abit the equation $dx+dy=1 \Rightarrow dx(1+\frac{dy}{dx})=1 \Rightarrow \frac{dy}{dx}=\frac{1}{dx}-1$.
We can see from this that the derivative dy/dx is not well defined because if it was then it would depend only on x or y or both but NOT on dx (or dy). This follows from the definition of derivative as the ratio of two differentials dy and dx.

 

[jonjacson]

thanks!