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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?

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?

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