Applying the Laplace transform to solve Differential equations

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Discussion Overview

The discussion revolves around the application of the Laplace transform to differential equations, particularly focusing on whether applying it to a finite order differential equation can yield an infinite order differential equation. Participants explore the implications and validity of this approach, as well as specific examples.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant questions the possibility of obtaining an infinite order differential equation from a finite order equation using the Laplace transform.
  • Another participant states that the Laplace transform converts a differential equation into an algebraic equation, but notes this is typically valid for equations with constant coefficients.
  • A different perspective suggests that expanding terms like ##\sin t\,y## as a series and applying the Laplace transform term by term could lead to derivatives of Y(s), though the validity and purpose of this method are uncertain.
  • A participant provides a specific example of the equation ##y''(t)+\sin(t)y(t)=0## and inquires whether applying the Laplace transform would result in an infinite order differential equation.

Areas of Agreement / Disagreement

Participants express differing views on the implications of applying the Laplace transform to finite order differential equations, with no consensus reached regarding the outcome of such applications.

Contextual Notes

There are limitations regarding the assumptions about the types of differential equations being discussed, particularly concerning the coefficients and the validity of series expansions in the context of the Laplace transform.

LagrangeEuler
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Is it possible to apply Laplace transform to some equation of finite order, second for instance, and get the differential equation of infinite order?
 
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A laplace transform turns a differential equation into an algebraic equation.
 
If you had a term like ##\sin t\,y##, you could expand ##\sin t## as a series and take the Laplace transform of the result term by term, which would give you a bunch of derivatives of Y(s). I'm not sure why you'd want to do that though or if doing so is valid.
 
pasmith said:
A laplace transform turns a differential equation into an algebraic equation.
It is only in the case when you have a differential equation with constant coefficients.
 
Do you have an example in mind?
 
##y''(t)+\sin(t)y(t)=0##, where ##y(0)=A##, ##y'(0)=B##. If I apply the Laplace transform would I get the differential equation of infinite order?
 

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