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Partial differentials (Need some reminders)

  1. Nov 1, 2012 #1
    It's been awhile since I've taken a differential equations course, so I just could not wrap my head around this one.

    1. The problem statement, all variables and given/known data
    I was given a lot of variables but it boils down to a partial differential equation that looks like:
    pT/pt = A*p^2T/px^2 + B*f(x)

    I am not looking for an answer to my particular problem, I want to understand how to go about solving it. I remember the basic differential problem form of eigenvalues---}general form---} particular solution for IVP's but I was given a problem with partials and my mind went blank!

    2. Relevant equations
    I wasn't sure where to start!


    3. The attempt at a solution
    Can you integrate with respect to a partial derivative?
     
  2. jcsd
  3. Nov 2, 2012 #2

    SammyS

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    Hello chrispsiler. Welcome to PF!

    Is that equation: ∂T/∂t = (A)(∂2T/∂x2) + (B)f(x) ?
     
  4. Nov 2, 2012 #3
    Yes, exactly. I was also given initial conditions of the form:
    T(0) for all x = k1
    T>0 , x=0 -c(∂T/∂2) = k2
    x=n , ∂T/∂x = k3
    where k1, k2, k3, n, and c are all given values.

    I tried treating it as a regular differential, but ran into a wall trying separation of variables and did not know how to think about it in my head. The given '-c(∂T/∂2) = k2' made me think that separation and integration were the way to go, but I could not separate the variables.

    It looks like the formula implies that a function of 'T' partially differentiated with respect to 't' equals the same function of 'T' partially differentiated twice with respect to 'x' added to a function of 'x'. What I don't understand is how you can find an analytical solution, because if something is partially differentiated with respect to one variable and to one other variable would not that allow the original function to be a function of infinitely many variables and still be expressible in that specific form? Thank you for any consideration or help, in advance!
     
  5. Nov 2, 2012 #4

    haruspex

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    Try writing T = U + h(x), where h satisfies Ah" + Bf = 0.
     
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