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Homework Help: ODE's for 2 space Heat equation

  1. Sep 2, 2010 #1
    1. The problem statement, all variables and given/known data
    The Heat equation in two space is

    [itex]\alpha ^2 \left[\frac{\partial ^2u}{\partial x^2}+\frac{\partial ^2u}{\partial y^2} \right]=\frac{\partial u}{\partial t}[/itex]

    Assuming separation solution of the form [itex]u(x,y,t)=F(x)G(y)H(t)[/itex] find ordinary differential equations satisfied by F,G and H.




    2. Relevant equations

    Heat Equation

    3. The attempt at a solution

    Because we can assume [itex]u(x,y,t)=F(x)G(y)H(t)[/itex]

    Is the first step that we require that


    [itex]\alpha^2\frac{F''}{F}+\alpha^2\frac{G''}{G}=\frac{H'}{H}[/itex]

    therefore we need to solve


    [itex]F''+\frac{k}{\alpha^2}F+G''+\frac{k}{\alpha^2}G=0[/itex]

    and

    [itex]H'-kH=0[/itex]


    Is this right so far?
     
  2. jcsd
  3. Sep 2, 2010 #2

    ehild

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    Homework Helper



    No. F"/F is a function of x, F"/F=f(x). Similarly, G"/G=g(y) and H"/H=h(t)

    α2(f(x)+g(y)) = h(t) holds only when f, g, h are all constant functions: f(x)=K, g(y)=L, h(t)=M

    with the condition that α2(K+L)=M
    Can you proceed from here?

    ehild
     
  4. Sep 2, 2010 #3
    Do I have to try to combine the first functions and set the right side to = 0 ?

    My examples in my notes are all of the form

    [itex]\alpha^2\frac{F''}{F}=\frac{ G'}{G}[/itex]
     
    Last edited: Sep 2, 2010
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