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Math Help - Autonomous system

  1. #1
    MHF Contributor alexmahone's Avatar
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    Autonomous system

    Consider the autonomous system

    \frac{dx}{dt}=F(x, y),\ \frac{dy}{dt}=G(x, y)

    in a region where the functions F and G are continuously differentiable.

    Suppose that (x(t), y(t)) is a solution of the system and that \gamma\neq 0. Define \phi(t)=x(t+\gamma) and \psi(t)=y(t+\gamma). Then show that (\phi(t), \psi(t)) is also a solution of the system.

    My attempt:

    (x(t), y(t)) is a solution of the autonomous system.

    So, x'(t)=F(x(t), y(t)) and y'(t)=G(x(t), y(t)).

    \frac{d\phi(t)}{dt}=x'(t+\gamma)

    \frac{d\psi(t)}{dt}=y'(t+\gamma)

    How do I proceed?
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: Autonomous system

    Quote Originally Posted by alexmahone View Post
    How do I proceed?
    \frac{d\phi(t)}{dt}=x'(t+\gamma)=F(x(t+\gamma),y(t  +\gamma))=F(\phi(t),\psi(t))

    \frac{d\psi(t)}{dt}=y'(t+\gamma)=G(x(t+\gamma),y(t  +\gamma))=G(\phi(t),\psi(t))
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  3. #3
    MHF Contributor alexmahone's Avatar
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    Re: Autonomous system

    Quote Originally Posted by FernandoRevilla View Post
    \frac{d\phi(t)}{dt}=x'(t+\gamma)=F(x(t+\gamma),y(t  +\gamma))=F(\phi(t),\psi(t))

    \frac{d\psi(t)}{dt}=y'(t+\gamma)=G(x(t+\gamma),y(t  +\gamma))=G(\phi(t),\psi(t))
    I'm not sure that I completely understand how x'(t+\gamma)=F(x(t+\gamma),y(t+\gamma)) and y'(t+\gamma)=G(x(t+\gamma),y(t+\gamma)).

    I know x'(t)=F(x(t),y(t)) and y'(t)=G(x(t),y(t)), but how do we know that substituting t+\gamma for t will lead to a valid equation?
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  4. #4
    MHF Contributor FernandoRevilla's Avatar
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    Re: Autonomous system

    For example, suppose x'(t)=F(x(t),y(t)) and y'(t)=G(x(t),y(t)) for all t\in\mathbb{R} . What is x'(u) and y'(u) for all u\in\mathbb{R} ?
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  5. #5
    MHF Contributor alexmahone's Avatar
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    Re: Autonomous system

    Quote Originally Posted by FernandoRevilla View Post
    For example, suppose x'(t)=F(x(t),y(t)) and y'(t)=G(x(t),y(t)) for all t\in\mathbb{R} . What is x'(u) and y'(u) for all u\in\mathbb{R} ?
    x'(u)=F(x(u),y(u)) and y'(u)=G(x(u),y(u)).
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  6. #6
    MHF Contributor FernandoRevilla's Avatar
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    Re: Autonomous system

    Quote Originally Posted by alexmahone View Post
    x'(u)=F(x(u),y(u)) and y'(u)=G(x(u),y(u)).
    Right, now use the substitution u=t+\gamma .
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  7. #7
    MHF Contributor alexmahone's Avatar
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    Re: Autonomous system

    Quote Originally Posted by FernandoRevilla View Post
    Right, now use the substitution u=t+\gamma .
    Hmm...makes sense. Thanks!
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