Differential equations

[wicked charm]

I think it's a way of writing differential equations of (n)-order

Such differential equation contains arguments such as

$\x, y, y'$, ... ,$y^{(n)}\$

[nova canyon]

adonhs

[wicked charm]

I think what's confusing you is that we think of this as y(x) a function with respect to one single variable.
But F(x,y,y',...) is something like a function of the differentials hence this differantial equation contains x, y, y'... etc.

[paper surge]

yes

is that

Why can I write it as a function of 3 variables? Yes, once I find "y" I will also find "y"

[wicked charm]

What do u mean exactly

[paper surge]

which is redundant, it is a function F(x,y, y') = 0 but once you have y(x) I can get everything else, I don't understand the point of writing it as 3 variables that in the end only depend on y.

sorry for the English, I speak Spanish and use a translator

[wicked charm]

F(x,y, y') refers to a differential equation of first order see it that way

this differentialequation contains an x, y and y'

[paper surge]

Ok, I understand that, if I see it like that it is a differential equation, but it does not solve my question about why it can be written like that

[wicked charm]

Math guys are lazy imagine you'd have this:

$\x^2y' + (x + ln(x) + e^x - x^x) * y + \frac{1}{x} = 0\$

You may aswell write it shorter:

$\F(x,y,y') = 0$

[nova canyon]

adonhs

[wicked charm]

Hmm this is a bad example

Maybe if you wanna do something proof related you can use this description F(x,y,y',..,y^n) = 0 to describe some differential equation of (n)-th order

[paper surge]

It is a good example, and I understand that it depends on 3 variables, what I say is that y' and y will only depend on y(x) so writing it like this I don't see the sense or the logic.

[wicked charm]

The same way you write:

f(x) = ln(x)
f(x,y) = xy
f(x,y,z) = z²y²x²

f(x,y) = xy could also be a differential equation like this:

f(x,y) = f(x,y(x)) = xy(x)

[wicked charm]

yeah y(x) refers only to x where as F(x,y,y',...) refers as a function of the whole differential equations

A function is like a machine that you can give inputs that gives outputs

[paper surge]

Ah okay so in a function you not only have to put the independent variables, but the dependent ones too?
For example
f(x,y) = xy
where y = f(x) = y(x)

[weak surge]

i think i see your confusion here @paper surge
you're asking why its F(x, y, y', y'' ... ) instead of just F(x, y), right?
well, i think it is so that you know to which degree the eq exists.
if you only write F(x, y), you wouldnt know if there is a y''' term, for example.
F(x, y, y', y'') makes it clear so that you only have those terms only. no y''' or y''''

[wicked charm]

Not really in F(x,y,y') this is a new function that refers to the whole differential equation. Your output F(x,y,y') of your differential equation depends of x,y,y' and so on

[weak surge]

+1

[paper surge]

Ok

ready

[weak surge]

ready? wut for? 😅

[paper surge]

I mean that's it

[weak surge]

ahh. alr alr

[wicked charm]

yeah that's pretty much middle school stuff very trivial

u got this

[paper surge]

How do we close the post?

[wicked charm]

.close

or .solved

[paper surge]

.close

[astral bronze]

Solved

Post marked as solved by @paper surge.

Use .unsolved if this was a mistake.