Comparison Of A Differential Equation To A Polynomial

Discussion in 'Math' started by Glenn Holland, Jan 14, 2015.

  1. Glenn Holland

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    Dec 26, 2014
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    Is there a layman's explanation of why a differential equation can be solved by converting the derivatives to exponents (making the "order" correspond to the "degree") and solving the resulting polynomial as the solutions to the differential equation?
     
  2. studiot

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    Are you referring to the D or differential operator method?
     
  3. WBahn

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    How familiar are you with Laplace and/or Fourier transforms?
     
  4. darrough

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    It would hard to imagine a layman's explanation of anything in differential equations. Its a tough class. :D
     
  5. Papabravo

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    The laymans explantion is that there is an isomorphism that is 1 to 1 and onto between the set of linear differential equations with constant coefficients and the set of polynomials with real coefficients. If you give me a polynomial that meets certain conditions I'll give you the corresponding differential equation. Conversely, if you give me a differential equation that meets certain conditions I'll give you the polynomial that leads to it's solutions.

    Differential equations and polynomials both have unique solutions. Isn't that special?
     
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  6. amilton542

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    I recognize this OP. He posts a question and never returns. I wouldn't waste your time.
     
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  7. WBahn

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    Oh well. I'm guessing that about 99% of our members that ever post anything make one post and never return. Ever.
     
  8. Glenn Holland

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    Most students study math for application to their immediate field, but they do not delve into the fine and subtle details of the subject. A carpenter goes to vocational school to learn how to work with wood, not to study the biology of trees.
     
  9. studiot

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    A carpenter who did not understand the effect of grain and the difference between the types of sawn planks would be in great difficulty when all his woodwork warped.

    Incidentally, why have you not replied to serious questions designed to help you with your question?
    Your original post was not clear exactly what you meant and different members have read it at least three different ways.
    It is up to you to let us know what you are after.
     
    Last edited: Jan 23, 2015
  10. WBahn

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    And most math classes intended for non-math majors do NOT delve into the fine and subtle details of the subject -- it just seems like that to the students who, pretty much across the board, do not realize the degree to which they are glossing over a lot of fundamental stuff related to the subject. Differential equations is typically a shining example of such a course.
     
  11. Glenn Holland

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    Dec 26, 2014
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    Actually, I was able to understand some of the finer details of math by experience with practical hands-on applications.

    I've been a tech geek and nerd since childhood and during my late teenage years, I frequently conversed with several scientists and engineers associated with the Caltech Seismological Lab in Pasadena. In their technical papers on earthquakes and instrumentation, they gave clear and practical implications/ramifications of the more subtle aspect of the mathematical analysis.

    However most students (at least the ones that I went to school with) were not geeks and nerds who habitually mingled with experts who could explain things from a practical viewpoint.
     
  12. WBahn

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    Practical experience can definitely help cement one's conception of mathematical concepts, but it seldom helps with the kind of fine points that mathematicians find important and that are often glossed over or left out entirely in treatments for non-mathematicians. For instance, I'm guessing that those papers seldom touched on things such as existence, convergence, the type of convergence, bounds, or uniqueness. And those are just some of the ones that I'm aware of as a non-mathematician.
     
  13. Glenn Holland

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    My observation is that in a capitalist and industrial economy, the primary objective of the engineering profession is the ability to use theoretical concepts to produce practical applications and ultimately marketable products.

    However the primary objective of the scientific profession is to produce theoretical concepts that may -or may not- have practical applications. Accordingly, engineering students are not required to have the strict knowledge of math that is frequently associated with the scientific profession.

    Many industrial economists also question whether the university and college system in the U.S. has an efficient blend of academic and practical education and the academic model from 50 years ago may not be valid in today's global economy.

    Market forces ultimately determine what is considered as a quality education. If wages and salaries are an indicator of what is considered as a quality education, it seems that community colleges and vocational schools are on the right track and universities may need to examine their objectives.
     
    Last edited: Jan 23, 2015
  14. darrough

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    I had an awful time time finding a job when I finished my bachelors degree in math. I spent several years working minimum wage jobs and wishing I had chosen a more practical major. But perhaps we are getting a bit off subject.
     
  15. darrough

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    I agree with you. A lot of the fine points are glossed over for engineering and science majors. There is nothing really wrong with that. Those majors have thier own fine points to worry about.
     
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  16. studiot

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    amilton you are correct, the OP clearly doesn't want an answer, he just wants to get into print.
     
  17. WBahn

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    Oh, I didn't say that there was anything wrong with it -- in fact it's probably a good thing and the way to do it. No one can learn every aspect of everything related to their profession at the level of detail needed to highly specialize in a given aspect. So while we need to settle for what's "good enough", we also need a way to determine what's "good enough" -- and not only is that a hard thing to do, but there are forces that are constantly trying to move the bar lower and lower and, because it's so hard to determine what's good enough we tend to slip into a realm where people are educated to a level that is not truly good enough long before that fact becomes evident.
     
  18. Glenn Holland

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    I've heard the saying about "Moving the bar lower" in a derogatory context, however it depends on what the objective of lowering the bar really is. In many cases, the objective is to get more people with practical skills instead of those with just purely academic qualifications.

    Many companies have voiced the opinion that recently graduated engineers lack a very basic knowledge of the field they studied. For example, many recent mechanical engineering grads could not pass a high school mechanical aptitude test such as knowing basic drafting, how to specify threaded fasteners, basic machining techniques, common material specs, and welding. They used to be called "Egg Heads", but today they're called "Theoretical Carpenters".

    With the manufacturing industry being more competitive, "Lowering the bar" may be a good thing.
     
  19. WBahn

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    So somehow we get people that know more just by expecting them to learn less?
     
  20. Glenn Holland

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    Industry wants to get people that know more of what the market requires rather than just what the academic community wants them to learn.

    The formal educational process is limited by the constraint on the time and money available. It's matter of "return on investment" and teaching/learning marketable skills takes priority over other areas that are perceived to have little or no financial value.
     
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