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Linear vs nonlinear diff equation II

  1. Oct 20, 2015 #1
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    For the linear differential equation, it says that a_n and b_n are constants or functions of x. This implies that a function of x = f(x) = constant. How can a number be a constant if it is a function? I think I'm misunderstanding something.
     

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  3. Oct 20, 2015 #2

    Geofleur

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    They mean that ## a_n ## and ## b_n ## can be either functions of ## x ## or constants, but not both. In other words, they are using "or" in the exclusive sense, not in the inclusive sense. We could also think of ## f(x) = ## constant as a special case of a "function of ##x##", albeit a trivial one. Think of a machine that spits out the same thing, no matter what you feed into it.
     
  4. Oct 20, 2015 #3
    If I am not mistaken, a trivial solution is one where the solution is zero?
     
  5. Oct 20, 2015 #4

    Geofleur

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    Well, trivial can also mean "not very interesting".
     
  6. Oct 20, 2015 #5

    SteamKing

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    Yes, the trivial solution is the zero solution.
     
  7. Oct 21, 2015 #6

    HallsofIvy

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    The crucial point in "linear" equations is that the coefficients, [itex]a_i[/itex], cannot be functions of y or any of its derivatives.
     
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