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Verifying if this PDE is a solution

  1. Feb 3, 2009 #1
    PROBLEM: Verify that the functions [x+1]e^(-t) ; e^(-2)sint ; and xt are respectively solutions of the nonhomogeneous equations
    Hu = -e^(-t)[x+1] ; Hu = e^(-2x)[4sint+cost] ; and Hu = x
    where H is the 1D heat operator H = [tex]\frac{\partial}{\partial t}[/tex] - [tex]\frac{\partial^2}{\partial x^2}[/tex]

    i did this the verification part,, the problem is with the second part of the problem
    Find a solution of the PDE
    Hu = [tex]\sqrt{2} x[/tex] + [Pi]e^(-2x) [4sint + cost] + e^(-t)[x+1]

    isn't the first part a solution for this PDE? i dont understand the question

    any hints how to setup this PDE?
  2. jcsd
  3. Feb 5, 2009 #2
    You have already verified some particular solutions for [tex]Hu =\phi(x,t).[/tex]
    Note that the DE is linear nonhomogeneous.

    If u1(x,t) (resp. u2(x,t)) is a solution of [tex]Hu =\phi_1(x,t)[/tex] (resp. [tex]Hu =\phi_2(x,t)[/tex] )
    u1(x,t) + u2(x,t) will be a solution of

    [tex]Hu =\phi_1(x,t) + \phi_2(x,t).[/tex]
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