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Partial derivatives of f(x,y)

  1. Apr 3, 2013 #1
    1. The problem statement, all variables and given/known data
    Where [itex]T(x,t)=T_{0}+T_{1}e^{-\lambda x}\sin(\omega t-\lambda x)[/itex]

    [itex]\omega = \frac{\Pi}{365}[/itex] and [itex]\lambda[/itex] is a positive constant.

    Show that T satisfies [itex]T_{t}=kT_{xx}[/itex] and determine [itex]\lambda[/itex] in terms of [itex]\omega[/itex] and k.

    I'm not to sure what is meant by the latter part of "determine [itex]\lambda[/itex] in terms of [itex]\omega[/itex] and k."


    2. Relevant equations



    3. The attempt at a solution

    So I think I first have to find the partial derivatives of the first order.
    [itex]\frac{\partial T}{\partial x}=-\lambda T_{1}e^{-\lambda x}\sin(\omega t-\lambda x)+T_{1}e^{-\lambda x}(-\lambda)\cos(\omega t-\lambda x)[/itex]

    [itex]\frac{\partial T}{\partial t}=T_{1}e^{-\lambda x}(\omega t)\cos(\omega t-\lambda x)[/itex]


    I then work out the second order partial derivative with respect to x and here it gets kind of messy and where I get confused.
    [tex]
    T_{xx}=-\lambda^{2}T_{1}e^{-\lambda x}\sin(\omega t-\lambda x)+T_{1}e^{-\lambda x}(-\lambda)\cos(\omega t-\lambda x)+(-\lambda^{2})T_{1}e^{-\lambda x}\cos(\omega t-\lambda x)+-\lambda^{2}T_{1}e^{-\lambda x}\sin(\omega t-\lambda x)[/tex]

    [tex]
    T_{xx}=-\lambda^{2}T_{1}e^{-\lambda x}\sin(\omega t-\lambda x)-\lambda^{2}T_{1}e^{-\lambda x}\sin(\omega t-\lambda x)-T_{1}e^{-\lambda x}\lambda\cos(\omega t-\lambda x)-\lambda^{2}T_{1}e^{-\lambda x}\cos(\omega t-\lambda x)[/tex]

    [tex]T_{xx}=-2\lambda^{2}T_{1}e^{-\lambda x}\sin(\omega t-\lambda x)-\lambda T_{1}e^{-\lambda x}\cos(\omega t-\lambda x)(1-\lambda)[/tex]

    [tex]T_{xx}=\lambda T_{1}e^{-\lambda x}(-2\lambda\sin(\omega t-\lambda x)-\cos(\omega t-\lambda x)(1-\lambda)
    [/tex]


    This looks nothing like the partial derivative of the first order with respect to t...
     
  2. jcsd
  3. Apr 3, 2013 #2

    phyzguy

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    Science Advisor

    You're just not doing the partial derivatives correctly. For example, when you take the partial derivative with respect to t, the multiplier should be ω, not (ωt). Also, in Txx, each term should be multiplied by λ^2. You missed a factor of λ in the second term. Also, don't forget that (-λ)^2 is equal to λ^2, not -λ^2.
     
  4. Apr 3, 2013 #3
    Ok I've improved it a little but still stuck...

    [tex]T_{xx}=\lambda^{2}T_{1}e^{-\lambda x}\sin(\omega t-\lambda x)+\lambda^{2}T_{1}e^{-\lambda x}\cos(\omega t-\lambda x)+(\lambda^{2})T_{1}e^{-\lambda x}\cos(\omega t-\lambda x)-\lambda^{2}T_{1}e^{-\lambda x}\sin(\omega t-\lambda x)[/tex]

    [tex]T_{xx}=\lambda^{2}T_{1}e^{-\lambda x}\sin(\omega t-\lambda x)-\lambda^{2}T_{1}e^{-\lambda x}\sin(\omega t-\lambda x)+\lambda^{2}T_{1}e^{-\lambda x}\cos(\omega t-\lambda x)+(\lambda^{2})T_{1}e^{-\lambda x}\cos(\omega t-\lambda x)[/tex]

    [tex]T_{xx}=\lambda^{2}T_{1}e^{-\lambda x}\cos(\omega t-\lambda x)+(\lambda^{2})T_{1}e^{-\lambda x}\cos(\omega t-\lambda x)=2\lambda^{2}T_{1}e^{-\lambda x}\cos(\omega t-\lambda x)[/tex]


    However, this still doesn't look right to me?

    Because if [itex]T_{t}=kT_{xx}[/itex]

    then

    [itex]\frac{\partial T}{\partial t}=T_{1}e^{-\lambda x}\omega\cos(\omega t-\lambda x)=kT_{xx}=k(2\lambda^{2})T_{1}e^{-\lambda x}\cos(\omega t-\lambda x)[/itex] which isn't true
     
  5. Apr 3, 2013 #4

    phyzguy

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    Science Advisor

    It is true for certain values of lambda. This basically gives you an equation which defines λ in terms of ω and k, which is what you were asked to provide.
     
  6. Apr 5, 2013 #5
    Ok I see, so it goes like this:

    Since[itex]\lambda[/itex]
    is a positive constant [itex]=kT_{xx}=k(2\lambda^{2})T_{1}e^{-\lambda x}\cos(\omega t-\lambda x)[/itex]
    and [itex]\frac{\partial T}{\partial t}=\omega T_{1}e^{-\lambda x}\cos(\omega t-\lambda x)[/itex]

    For [itex]\frac{\partial T}{\partial t}=kT_{xx}[/itex] to be true [itex] k(2\lambda^{2})=\omega[/itex] therefore [itex]\lambda=\sqrt{\frac{\omega}{2k}}[/itex]


    Then [itex]T_{t}=kT_{xx}[/itex]
    only when [itex]\lambda=\sqrt{\frac{\omega}{2k}}[/itex]
     
  7. Apr 5, 2013 #6

    phyzguy

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    Science Advisor

    You got it!
     
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