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Differential equation with only the trivial solution

  1. Dec 27, 2012 #1
    1. The problem statement, all variables and given/known data
    Find a differential equation with its only (complex-valued) solution being y=0


    2. Relevant equations



    3. The attempt at a solution
    I believe that there is no DE having only y=0 as its solution, but frankly I am not sure if this is the case. I would like to know whether or not this is true, so that I know in which direction I can begin working my proof (or at least a hint is appreciated).

    Also, something (in my mind) tells me this problem may have some connection with Wronskian matrices, but I have no clue really.

    EDIT: I tried playing with some random diff Eqs, it seems that there are in fact trivial things like y+y'-y'=0, but this type of answer seems to be trivial to be of substance. Is there a DE that does not reduce to a trivial y=0?

    Thanks!

    BiP
     
  2. jcsd
  3. Dec 27, 2012 #2

    haruspex

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    One approach is to think of a function of y' that cannot be zero, f(y') say, then write f(y')y = 0.
     
  4. Dec 27, 2012 #3

    Dick

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    Well, f(y')=1 works. Still seems like kind of a cheat. Notice they also said complex valued. So something like f(y')=1+(y')^2 won't work either. Not to say you can't cook one up. Like f(y')=(1+(y')(y'*)). I'm just wondering if there is something nontrivial here.
     
    Last edited: Dec 27, 2012
  5. Dec 28, 2012 #4

    haruspex

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    I was thinking of ey'y=0
     
  6. Dec 28, 2012 #5
    Good work!!! And thanks!!!

    BiP
     
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