Can Integration Generate a Characteristic Equation?

  • Thread starter Jhenrique
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In summary: However, the definition of the harmonic derived is obscure to me and I'm not sure what you are trying to achieve.
  • #1
Jhenrique
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4
If from the derivate, we can generate an equation that is the equation of the tangent straight, so:

[tex]\frac{\mathrm{d} y}{\mathrm{d} x}=\frac{\mathrm{d} y}{\mathrm{d} x}[/tex]
[tex]\mathrm{d} y=\frac{\mathrm{d} y}{\mathrm{d} x}\mathrm{d} x[/tex]
[tex]y=\frac{\mathrm{d} y}{\mathrm{d} x}\mathrm{d} x+y_{0}[/tex]
[tex]y(x)=y'(x_0)(x-x_0)+y(x_0)[/tex]

And this extends even to other cases...

[tex]y(x)=y''(x_0)\frac{(x-x_0)^2}{2}+y'(x_0)(x-x_0)+y(x_0)[/tex]
[tex]y(x)=y^{**}(x_0)^{\frac{(x-x_0)^2}{2}}\times y^{*}(x_0)^{(x-x_0)}\times y(x_0)[/tex]

Being
[tex]y^{*}(x)=exp\frac{f'(x)}{f(x)}[/tex]
The geometric derivate

... So, similarly, is not possible to generate a characteristic equation with integration?
 
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  • #2
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.
 
  • #3
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...
 
  • #4
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.
 
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  • #5
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?
 
  • #6
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.
 

1. What is an Integral's equation?

An Integral's equation is an equation that represents the relationship between a function and its integral. It involves the use of an integral sign (∫) and is commonly used in calculus to find the area under a curve.

2. How is an Integral's equation solved?

An Integral's equation can be solved using various methods such as the substitution method, integration by parts, and partial fractions. These methods involve manipulating the equation in order to isolate the integral and then using integration rules to evaluate it.

3. What is the purpose of an Integral's equation?

The purpose of an Integral's equation is to find the value of a function's integral, which represents the area under the curve. This is useful in many applications, such as finding the displacement of an object with varying velocity or calculating the work done by a variable force.

4. Can an Integral's equation be used for any function?

Yes, an Integral's equation can be used for any continuous function. However, some functions may be more difficult to integrate than others and may require more advanced techniques to solve the equation.

5. Are there any limitations to using an Integral's equation?

One limitation of using an Integral's equation is that it can only be used for continuous functions. It also may not be applicable in some cases where the function is not well-defined or does not have a definite integral. Additionally, the accuracy of the solution depends on the accuracy of the given function and the integration method used.

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