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Math Help - Is this ODE linear?

  1. #1
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    Is this ODE linear?

    If I have the ODE:

    x`=x2*et+ex

    can this differential equation be considered linear?

    I know that the definition of a linear ODE is that it can be written as:
    x`+a(t)x=b(t)

    if I take the sqrt of my initial ODE than I have:
    x'-(et)(1/2)*x=(ex)(1/2)

    This would be a linear differential equation with:
    a(t)=(et)(1/2) and
    b(t)=(ex)(1/2)

    is this true?
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  2. #2
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    Re: Is this ODE linear?

    Quote Originally Posted by infernalmich View Post
    If I have the ODE:

    x`=x2*et+ex

    can this differential equation be considered linear?

    I know that the definition of a linear ODE is that it can be written as:
    x`+a(t)x=b(t)

    if I take the sqrt of my initial ODE than I have:
    x'-(et)(1/2)*x=(ex)(1/2)

    This would be a linear differential equation with:
    a(t)=(et)(1/2) and
    b(t)=(ex)(1/2)

    is this true?
    No, it is not true "b(t)= (ex)(1/2)" is impossible because the right side is a function of x, not t.

    If you thinking, "It is a function of t because x is a function of t", that is true but not the point here. In deciding whether a differential equation is linear or non-linear we treat the variables as if they were independent.
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  3. #3
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    Re: Is this ODE linear?

    ok, now I understand!

    Thank you for your help. But if it would be (et)(1/2) instead of (ex)(1/2) it would be linear, right?
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