What are the general forms of implicit and parametric equations?

[Jhenrique]

I was thinking about some form of classify all kinds of equations and system of equations. And the better classification that I found is by 1st to classify if the equation represents a curve or a surface. Represents a curve if the spatial coordinates are functions of an unique variable and surface if are functions of two variables. Until here, ok. So, exist equations, system of equations, differential equations and system of differential equations (I'll limit myself to 2D only), but what is the general form of all?

So I realize that an implicit equation/function is $F(x,y) = F(x,f(x)) = 0$ (obvius), and its explicit form is $y = f(x)$. But exist too the parametric equations, which is its explicit and inplicit general form? This is a good question! Its implicit general form is $F(x,y,t) = F(x(t),y(t),t) = F(\vec{r}(t),t) = 0$ and the explicit form is $\vec{r} = \vec{r}(t)$.

Its differential form, respectively, are: $F(x,y,\frac{dy}{dx}) = 0$, $\frac{dy}{dx}=f(x,y)$, $F(t, \vec{r}, \frac{d\vec{r}}{dt}) = 0$ and $\frac{d\vec{r}}{dt}=\vec{f}(t, \vec{r})$. About this last, would be that it is autonomus? Although it is function of $t$ too, $t$ isn't a spatial coordinate, ie, $t$ isn't ploted in the plane xy...

The solution of a system of equations is to find a point that satisfies 2 equations, ie, to find the intersection between 2 curves.

It is said that $\frac{d\vec{r}}{dt} = A \vec{r}$ is a system of differential equations, but I don't think so, IMO this equations represents a parametric differential curve, and, by definition, is necessary another equation (another curve) for to find the intersections between those curves.

Ie, nobody until now studied, in actually, a system of differential equations! You already realized this?

 

[Simon Bridge]

...the better classification that I found

... is this some personal theory you have of your own or are you asking about how mathematicians classify equations?

So, exist equations, system of equations, differential equations and system of differential equations (I'll limit myself to 2D only), but what is the general form of all?

... every possible differential equations imaginable? I'll answer this question with another question: what is the general form for an arbitrary 2-variable equation?

Once you've got that - expand it to include arbitrary combinations of any order of mixed partials.

IMO this equations represents a parametric differential curve...

Your opinion on this matter is incorrect.
$\frac{d}{dt}\vec r = A\vec r$ represents a system of linear DEs to 1st order.
These things may have solutions without resorting to an additional equation.
This is easy to prove.

Ie, nobody until now studied, in actually, a system of differential equations!

This is not correct - systems of DEs forms part of a standard college calculus course.

 

[Jhenrique]

Just because you have a differential equation, or a algebraic equation, for x and for y not means that you have a system of equations. dr/dt = Ar have a diff eq for x and y and describe only an unique curve. Like I said, a system of equations in 2D have 2 curve (2 parametric/direct/implicit/explicit curve, whatever) and the solution for the system is a point of intersection between the curves.

Realize that dr/dt = Ar is the analogous of dy/dx = ay, those 2 equations represents a curve, the difference is that one have the coordinates as function of a paramater and the other have a coordinate function of other coordinate.

An system of differential equations is:
$F_1(x,y,\frac{dy}{dx}) = 0$
$F_2(x,y,\frac{dy}{dx}) = 0$

or

$F_1(t, \vec{r}, \frac{d\vec{r}}{dt}) = 0$
$F_2(t, \vec{r}, \frac{d\vec{r}}{dt}) = 0$

 

[Simon Bridge]

Just because you have a differential equation, or a algebraic equation, for x and for y not means that you have a system of equations.

So what?

If you have more than one equation of any kind does not mean you have a system of simultaneous equations - you also need the assertion that all the equations must be true simultaneously.

The same holds for systems of differential equations.

So sure: if you insist - the relation: $$\frac{d}{dt}\vec r = A\vec r$$ ... only represents a system of DEs if someone says that it does.

Nothing new or profound about this - just normal maths.

Your examples are more general, including the possibility that the differential equations can be non-linear.
If F1 is x'=ax+by and F2 is y'=cx+dy then the system can be written in the previous form using a 2x2 matrix.
Other systems would have different representations.

Check the mathematical definition of "a system of differential equations" and you will find that it is a matter of routine study.

So what are you trying to say?
Do you believe you have discovered something new?

 

[Mark44]

As Simon has already said, systems of differential equations, and in particular, systems of linear differential equations are already part of the curriculum for most college courses on differential equations.

Putting aside the "systems of" part, ordinary differential equations are already divided into a number of classes: linear, nonlinear, homogeneous (with two different meanings for the term "homogeneous"), nonhomogeneous, first order, second order, third order, and quite a few others. What advantage does your classification scheme have over the classifications that are already in existence?

English lesson:
Your thread title is "No exist system of diff eq". This might be a direct translation of something like "no existe una sistema de ..." or maybe "no existe uma sistema do..." I don't speak Portuguese, but I speak some Spanish, so I'm pretty sure what I wrote is understandable to you.

The question, "¿No existe uma sistema do ...?" would be phrased in English as "Does there exist a system of ...?" If your title isn't a question, but is instead a statement, it would be phrased in English as "There does not exist a system of ..." or "A system of ... does not exist."

 

[Jhenrique]

Yeah, my title is a declaration for cause impact!

The advantage of my classification (isn't well a classification, is just a observation) is that a curve can be describied or by a explicit function, or by a implicit function or by a explicit parametric vector or by a implicit parametric vector. And each one those types have a form differential.
The second ideia is that a curve isn't system of equation, a system of equation in 2D is constituted by 2 curves.
Second this point of view, no exist none study of system of differential equations, ie, no exist none theory about how to find the solution for 2 differential curves simultaneously, ie, to find a point that belongs to 2 differential curves.

Do you understand what I want say?

 

[micromass]

So there isn't really a question here. You just propose some definitions that nobody else uses. This is getting close to breaking our rules.

Thread locked.