Can Integration Generate a Characteristic Equation?

[Jhenrique]

If from the derivate, we can generate an equation that is the equation of the tangent straight, so:

$$\frac{\mathrm{d} y}{\mathrm{d} x}=\frac{\mathrm{d} y}{\mathrm{d} x}$$
$$\mathrm{d} y=\frac{\mathrm{d} y}{\mathrm{d} x}\mathrm{d} x$$
$$y=\frac{\mathrm{d} y}{\mathrm{d} x}\mathrm{d} x+y_{0}$$
$$y(x)=y'(x_0)(x-x_0)+y(x_0)$$

And this extends even to other cases...

$$y(x)=y''(x_0)\frac{(x-x_0)^2}{2}+y'(x_0)(x-x_0)+y(x_0)$$
$$y(x)=y^{**}(x_0)^{\frac{(x-x_0)^2}{2}}\times y^{*}(x_0)^{(x-x_0)}\times y(x_0)$$

Being
$$y^{*}(x)=exp\frac{f'(x)}{f(x)}$$
The geometric derivate

... So, similarly, is not possible to generate a characteristic equation with integration?

 

[Simon Bridge]

The fifth line is basically the Taylor series about $x_0$ - I don't know what you are trying to achieve though.
You can generate a characteristic equation by many means - but "characteristic" of what?
That's the important part.

 

[Jhenrique]

These equations have important meaning! The equation (4) serves to show the instantaneous rate of change of function, that serves to visualize the derivative of the function. Equation (5) is a parable and serves to show the concavity of the graph, in the inflection points, the parabola degenerates and becomes a straight line. Equation (6) is a parabola in log-normal plane and performs the same job as the (5).

If the derivatives are able to generate equations that reveal information about the graph, the integral, maybe, may generate some kind of equation that reveal other information.

Maybe not the integral, maybe yes the harmonic derived can to generate other interesting equation, but the definition of harmonic derivate is obscure to me...

 

[Simon Bridge]

If the derivatives are able to generate equations that reveal information about the graph, the integral, maybe, may generate some kind of equation that reveal other information.

Integration and differentiation are inverse processes - each will reveal new levels of information about the relationship represented by the graph. Graphs have no meaning by themselves.

i.e. if you keep track of the velocity of an object at different times, you can graph that as v(t) vs t.
The first (time) derivative v(t) tells you the instantaneous acceleration and integral tells you the displacement.

Integrations can, indeed, be used to generate other interesting equations.

If integrations did not reveal reveal information, they would not be useful.

... the definition of harmonic derivate is obscure to me...

... I'm sorry, please ask a question.

 

[Jhenrique]

I know! I'll rephrase my question ... Do you have idea how I could continue the taylor's series in the direction opposite to the conventional? See equation (4) exist some way of continue to expand it to the right side?

 

[Simon Bridge]

No - the Taylor series only goes forward.

When you differentiate you lose information - the arbitrary constant on integration - so there is no unique way to go the other way.

Technically you could go backwards in some series you invent that uses integration rather than differentiation - keeping the arbitrary constant as an additional parameter- but, then, the expression gets more and more vague as you add terms so it is self-defeating.

There is a way you can create a characteristic equation by integration - that is the process of solving differential equations.