policer

What is the name of the thing that mark with ...?
mx + n = y
n is ...
ax^2 + bx + c = y
c is ...

policer

And my next question is:
What is the difference between this Thing and the Constant Of Integration?

skipjack

The technical term is "ellipsis". It isn't related to integration.

Micrm@ss

I think he is refering to the "constant term".

policer

What is the name of The Thing?
mx + n = y
n = The Thing
ax^2 + bx + c = y
c = The Thing.
this is a "free number" in my bad English.
If integration of f function is
(Integral of f) + C
what is the difference between c and the "free number"?
Is there a connection between them?

skipjack

In your equations, n and c appear to be constant terms, as there is nothing to suggest they vary as x varies. When you integrate with respect to x, the C that is added as a constant of integration doesn't vary as x varies, but (in certain circumstances) might vary as something else varies. I haven't come across the term "free number" in this type of context.

policer

What are the circumstances?

Country Boy

If you have a "partial differential equation" that has more than one independent variable is one such circumstance.

For example, to solve the differential equation $\displaystyle \frac{\partial^2 z}{\partial x\partial y}= 0$, write it as $\displaystyle \frac{\partial}{\partial x}\left(\frac{\partial z}{\partial y}\right)= 0$. The "anti-derivative", with respect to x, would give $\displaystyle \frac{\partial z}{\partial y}$ plus a "constant". But since this is a partial derivative with respect to x, where we hold y constant, that "constant might in fact be some function of y! That is $\displaystyle \frac{\partial z}{\partial y}+ f(y)$. Now integrating with respect to y, we have $\displaystyle z= F(y)+ "C"$ where F is an anti-derivative of f. And, again, since we differentiated with respect to y, that "C" might be some function of x. If we call that function "G(x)", that becomes $\displaystyle z= F(y)+ G(x)$.