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Math Help - Linear/non-linear

  1. #1
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    Linear/non-linear

    Just a simple question:

    Is cosxdy/dx = x non-linear because of cosx?
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by CookieC View Post
    Just a simple question:

    Is cosxdy/dx = x non-linear because of cosx?
    An differential operator $$ L is linear iff for any two solutions y_1(x) and y_2(x) of Ly=0 and any constants \alpha, \beta \in \mathbb{C} then y(x)=\alpha y_1(x)+\beta y_2(x) is also a solution. An ODE is linear if it is of the form Ly=f(x),\ L a linear differential operator

    Now you can check for yourself.

    (the answer is: it's linear)

    CB
    Last edited by CaptainBlack; September 14th 2010 at 04:06 PM. Reason: fix a dreadful error
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  3. #3
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    Any first ODE that can be written as

    \dfrac{dy}{dx} = P(x) y + Q(x)

    i.e. linear in both y and y' is a linear ODE. Your ODE is

    \dfrac{dy}{dx} = \sec x \, y so it's \cdots
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  4. #4
    Grand Panjandrum
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    Quote Originally Posted by Danny View Post
    Any first ODE that can be written as

    \dfrac{dy}{dx} = P(x) y + Q(x)

    i.e. linear in both y and y' is a linear ODE. Your ODE is

    \dfrac{dy}{dx} = \sec x \, y so it's \cdots
    I would rather stick with the definition of linearity; less to remember. It is also a property of interest (that solutions satisfy superposition)

    CB
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  5. #5
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    But aren't you assuming two independent solutions (for a first order ODE?).
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  6. #6
    Grand Panjandrum
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    Quote Originally Posted by Danny View Post
    But aren't you assuming two independent solutions (for a first order ODE?).
    As it happens, no. The definition still holds. (Alternatively, what else would linear mean?)

    CB
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  7. #7
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    According to the wiki, a linear differential equation is of the form

    Ly(x)=f(x), where L is a linear differential operator.

    That is, you apply CB's definition of linearity to the operator, not necessarily to the entire equation. If the equation is homogeneous, they'll turn out to be the same thing. But even a non-homogeneous linear DE does not obey the superposition principle except in its homogeneous solution.

    This is my understanding.
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  8. #8
    Grand Panjandrum
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    Quote Originally Posted by Ackbeet View Post
    According to the wiki, a linear differential equation is of the form

    Ly(x)=f(x), where L is a linear differential operator.

    That is, you apply CB's definition of linearity to the operator, not necessarily to the entire equation. If the equation is homogeneous, they'll turn out to be the same thing. But even a non-homogeneous linear DE does not obey the superposition principle except in its homogeneous solution.

    This is my understanding.
    Opps.. that is what I should have said.

    CB
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