From integral equation to normal equation

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Discussion Overview

The discussion revolves around the transition from an integral equation to a normal equation in the context of differential equations. Participants are examining specific equations and their correctness, exploring methods such as the Laplace transform, and questioning the relationships between derivatives and their implications.

Discussion Character

  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Post 1 questions the correctness of two equations derived from a second-order differential equation and seeks clarification on the transition to a normal equation.
  • Some participants suggest using the Laplace transform as a method to solve the differential equation presented in Post 1.
  • Post 3 reiterates the use of the Laplace transform but asserts that the original equation is not a differential equation.
  • Post 4 challenges the validity of equation [1] by pointing out that it does not logically follow from the initial equation [0] and notes the presence of a linear term in the right-hand side.
  • Post 5 provides a modified version of equation [1], indicating an attempt to clarify or correct it.
  • Post 6 echoes the concern raised in Post 4 about the correctness of equation [1] and requests further explanation regarding the derivation of "x".

Areas of Agreement / Disagreement

Participants express disagreement regarding the correctness of equation [1] and its derivation from equation [0]. There is no consensus on the validity of the equations or the methods proposed for transitioning between them.

Contextual Notes

Some participants highlight that the original equation may not be a differential equation, which introduces ambiguity in the discussion. The relationship between the equations and the assumptions made about the variables is also under scrutiny.

ManishR
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Consider

\frac{d^{2}y}{dx^{2}}=k--------------[0]

and

\frac{dy}{dx}=y_{a}

Then

y_{f}-y_{i}=\frac{k}{2}(x_{f}^{2}-x_{i}^{2})-----------[1]

y_{af}-y_{ai}=k(x_{f}-x_{i})-------------[2]

is equation 1 and 2 correct ? if no, then what is the correct solution

if yes

\Rightarrow\frac{d^{2}y}{dx^{2}}=\frac{dy}{dx}=(positive/negative)

or more correctly

\Rightarrow\frac{\frac{d^{2}y}{dx^{2}}}{\left|\frac{d^{2}y}{dx^{2}}\right|}=\frac{\frac{dy}{dx}}{\left|\frac{dy}{dx}\right|}=\frac{k}{\left|k\right|}

assumming

x_{f}>x_{i}

but that's not always true for example its not true for this

y=2x-x^{2}-------------------[4]

where

-1\leq x\leq1

\Rightarrow\frac{dy}{dx}=2-2x=positive

\Rightarrow\frac{d^{2}y}{dx^{2}}=-2=negative

so how to derive from integral equation (like [0]) to normal equation(like [4]) ?
 
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Last edited by a moderator:
jjmontero9 said:
This can be done using the http://en.wikipedia.org/wiki/Laplace_transform" , which is really useful for solving differential equations. First you apply the direct transformation to [0], then solve for y and then apply the inverse laplace transform to find an equation like [4].

An example on http://en.wikipedia.org/wiki/Laplace_transform#Example_.231:_Solving_a_differential_equation"

Its not a differential equation.
\frac{dk}{dx}=\frac{dk}{dy}=0
 
Last edited by a moderator:
Your [1] doesn't really follow from [0]. There's also a linear term in "x" in the RHS.
 
thanks bigubau
y_{f}-y_{i}=y_{ai}(x_{f}-x_{i})+\frac{k}{2}(x_{f}^{2}-x_{i}^{2}) -------[1]
 
bigubau said:
Your [1] doesn't really follow from [0]. There's also a linear term in "x" in the RHS.

could you explain how you got x ?
i still cannot see how equation [1] is wrong ?
 

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