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Homework Help: Let v(x,t) = u(x+ct) and show that

  1. Mar 19, 2012 #1
    1. The problem statement, all variables and given/known data

    NOTE: I realize the following is a partial differential equation, but I believe the answer to my question is a straight forward multi-variate calculus question.

    Let v(x,t) = u(x+ct,t) and show that v(x,t) solves the following...

    ut = κ uxx ; (-∞<x<∞)

    u(x,0) = [itex]\phi[/itex](x)

    NOTE: You may disregard the following, if necessary for you to answer this question:

    u(x,0) = [itex]\phi[/itex](x)

    and simply show that v(x,t) solves the following...

    ut = κ uxx

    ALTERNATIVELY, you may help me by commenting on the correctness of my work below.

    2. Relevant equations

    See above and below...

    3. The attempt at a solution

    v(x,t) = u(x+ct,t)

    v'(x,t) = c u[itex]_{x}[/itex](x+ct,t) + u[itex]_{t}[/itex](x+ct,t)

    v[itex]_{x}[/itex](x,t) = c u[itex]_{x}[/itex](x+ct,t)

    v[itex]_{xx}[/itex](x,t) = c[itex]^{2}[/itex] u[itex]_{xx}[/itex](x+ct,t)

    v[itex]_{t}[/itex](x,t) = u[itex]_{t}[/itex](x+ct,t)
  2. jcsd
  3. Mar 19, 2012 #2
    Any help on any part will be GREATLY appreciated!!!
  4. Mar 19, 2012 #3


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    Gold Member

    Welcome to PF,

    Just before we start, are you sure it was not meant to be simply v(x,t) = u(x+ct)? Because I know that a function in which the temporal and spatial dependence are related in this way will satisfy a wave equation, which is given by utt = κuxx. This is different from the PDE that you gave. But anyway, taking what you wrote at face value:

    First of all, I don't think that v' (v "prime") is meaningful here. In single-variable calculus, a prime symbol typically means "derivative with respect to the argument." But this function has two arguments. Since it is a multi-variable function, any derivative is going to be a partial derivative, and you must explicitly specify which variable you are differentiating with respect to. As for your attempt at partial derivatives, my approach would be to write:

    u = u(w,t) where w = x+ct


    [tex]\frac{\partial u}{\partial x} = \frac{\partial u}{\partial w}\frac{\partial w}{\partial x}[/tex][tex]= \frac{\partial u}{\partial w}[/tex]

    Et cetera for the other derivatives
  5. Mar 19, 2012 #4
    To your first point...

    I am sure it is:

    Let v(x,t) = u(x+ct,t)

    to your second point...

    That would mean the following is incorrect, right?

    v.sub.x(x,t) = c u.sub.x(x+ct,t)

    and should have been...

    v.sub.x(x,t) = u.sub.x(x+ct,t)
  6. Mar 19, 2012 #5


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    What I was saying was that v_x = u_w, with underscores denoting subscripts.
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