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Linear or Nonlinear Differential Equation?

  1. Aug 31, 2012 #1
    1. The problem statement, all variables and given/known data

    Is this differential equation linear or nonlinear? Assume that y' means dy/dx.

    2. Relevant equations

    1. The problem statement, all variables and given/known data[/b]

    Is this differential equation linear or nonlinear? Assume that y' means dy/dx.

    2. Relevant equations

    [itex]\sqrt{xy'+2x2}[/itex]=5

    3. The attempt at a solution

    It seems to me that squaring both sides and subtracting 2x2 from both sides renders a linear equation:

    xy' + 2x2=25.

    xy' = 25 - 2x2

    This is of the form

    an(x)y(n) + an-1(x)y(n-1) + ... a0(x)y=g(x)

    if you take a0 = 0... isn't it?

    The problem is that my professor says otherwise; namely, that it is a nonlinear DE.

    Where am I making my mistake?

    Thanks.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Aug 31, 2012 #2

    Bacle2

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    The way I know it, in order to decide if the operator is linear,you express it (the

    differential operator) in the form f(y) and then see if f is linear in that sense, i.e.,

    if h is another function, is f(y+h)=f(y)+f(h) ,is f(cy)=cf(y).
     
  4. Aug 31, 2012 #3

    SammyS

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    WolframAlpha characterizes it as a first-order linear ordinary differential equation.
     
  5. Aug 31, 2012 #4
    Where on Wolfram Alpha are you able to find something that tells you whether a DE is linear or nonlinear?
     
  6. Aug 31, 2012 #5

    LCKurtz

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    I don't think you will find unanimity on this question. Some sources say a linear DE is one that has the form ##a_n(x)y^{(n)} + a_{n-1}(x)y^{(n-1)} + ... a_0(x)y=g(x)## while others any DE that can be put in that form is linear. It depends on the definition your professor gave.
     
  7. Aug 31, 2012 #6

    Bacle2

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    But I still believe the operator definition holds. This agrees with your definition, Mr

    LCKurtz : yourform is linear on the function y.
     
  8. Aug 31, 2012 #7

    LCKurtz

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    But the question was about this DE:

    ##\sqrt{xy'+2x^2}=5##

    Certainly the left side is not linear in y' as it stands, although it can be put in the linear form by squaring.
     
  9. Sep 5, 2012 #8
    My professor gave the "can be put in linear form" definition; hence my confusion. Thank you guys for easing my mind.
     
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