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Math Help - What makes an ODE linear?

  1. #1
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    What makes an ODE linear?

    I'm having trouble understanding this for some reason...am I right about these?

    xy' - 4y = 0 is linear.

    y' = y - x^3 is linear.

    x' + 2x - e^(-x) = 0 is non-linear.
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  2. #2
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    Re: What makes an ODE linear?

    Quote Originally Posted by euphony View Post
    I'm having trouble understanding this for some reason...am I right about these?

    xy' - 4y = 0 is linear.

    y' = y - x^3 is linear.

    x' + 2x - e^(-x) = 0 is non-linear.
    An ODE is linear if it is of the form a(x)y^{(n)} + b(x)y^{(n - 1)} + ~...~ + c(x)y = F(x). Where y^{(n)} is the nth derivative of y(x) and all the a(x), b(x), ...., c(x), F(x) are all functions of x only.

    -Dan
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  3. #3
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    Re: What makes an ODE linear?

    Note that in the first two you are thinking of "x" as the independent variable- y is a function of x. But in the last, x is the dependent variable- x is a function of some unnamed variable.
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