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Thread: Laplace Transformation vs. Slope field

  1. #1 Laplace Transformation vs. Slope field 
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    Which method do physicists use to solve differential equations more: slope fields or Laplace transformations? I was under the impression that an applied physicist would use the latter more, while a theoretical physicist would be more likely to use the former.


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  3. #2 Re: Laplace Transformation vs. Slope field 
    . DrRocket's Avatar
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    Quote Originally Posted by Ellatha
    Which method do physicists use to solve differential equations more: slope fields or Laplace transformations? I was under the impression that an applied physicist would use the latter more, while a theoretical physicist would be more likely to use the former.
    It depends on the problem.

    Laplace transforms are generally used for initial value problems given an ordinary differential equation with constant coefficients. This application is very common in electrical engineering. This method produces exact solutions and is capable of describing the solutions to an entire family of closely related differential equations. So, for instance, one can develop a theory of stability around the Laplace transform if the system is governed by ordinary differential equations with constant coefficients, not necessarily of first order. This is important in control theory.

    Slope fields provide a graphical, hence numerical, approximation to a solution of a first-order ordinary differential equation, not necessarily with constant coefficients. They have the advantage of providing a graph of a solution, even if the solution is not describable in closed form. They have the disadvantage of not providing a solution in closed form even if one exists.

    The plain truth is that a great many interesting problems in physics and engineering are not described by first-order ordinary differential equations of one variable. So, in practice other methods are often needed.

    There are all sorts of techniques for handling these problems. The theory of ordinary and partial differential equations is a big subject.


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