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Using change of variables to change PDE to form with no second order derivatives

  1. Feb 21, 2012 #1
    1. The problem statement, all variables and given/known data
    Classify the equation and use the change of variables to change the equation to the form with no mixed second order derivative. u_{xx}+6u_{xy}+5u{yy}-4u{x}+2u=0



    2. Relevant equations

    I know that it's of the hyperbollic form by equation a_{12}^2 - a_{11}*a_{22}, which gives 4 >0 so it's hyperbollic.

    3. The attempt at a solution
    What I don't understand is how to get started on changing the form to something with no second order derivatives. My book used a matrix method but shows no examples. Could I have a tip for where to get started? Thanks!
     
  2. jcsd
  3. Feb 21, 2012 #2

    HallsofIvy

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    You don't want "no second order derivatives", you want "no second order mixed derivatives". In other words, you want to get rid of that "[itex]u_{xy}[/itex]" term.

    Think of this as [itex]x^2+ 6xy+ 5y^2[/itex], a quadratic polynomial.

    Rotating through an angle [itex]\theta[/itex] would give [itex]x= x'cos(\theta)- y'sin(\theta)[/itex], [itex]y= x'sin(\theta)+ y'cos(\theta)[/itex]. Put those into the polynomial and choose [itex]\theta[/itex] so that the x'y' term is 0.

    Equivalently, simpler calculations but more sophisticated, write the polynomial as
    [tex]\begin{bmatrix}x & y\end{bmatrix}\begin{bmatrix}1 & 3 \\ 3 & 5\end{bmatrix}\begin{bmatrix}x \\ y\end{bmatrix}[/tex]
    Use the eigenvectors of that matrix as the new axes.
     
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