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Math Help - second order partial differential equation

  1. #1
    tur
    tur is offline
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    second order partial differential equation

    Can someone help me to solve this equation or even classify it..

    (z^a)*p[x,y,z]=(x^2)*p[x,y,z]-(2*x*y)*p'[x,y,z]+(y^2)*p''[x,y,z]

    where p'[x,y,z],p''[x,y,z] means partial derivative with respect to z

    and 0<a<1

    Thanks
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  2. #2
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    Here's my suspicion but I'm not entirely sure: Don't let the partials intimidate you. It's still an ordinary DE in the variable z:

    y^2\frac{\partial^2 p}{\partial z^2}-(z^a+2xy)\frac{\partial p}{\partial z}+x^2u=0

    Alright let's down-grade it first: drop the a for now and consider just the ordinary-looking DE:

    ay''-(b+x)y'+cy=0;\quad y=f(x)

    For constants a, b, and c. See the similarity? Ok, we can probably solve this and end up with a solution containing two arbitrary constants along with the a,b, and c (I'll call the solution p now):

    p(x;\{a,b,c,k_1,k_2\})

    Now, in terms of the PDE, the arbitrary constants are functions of the other two variables. Call them f(x,y), g(x,y). Now, using the solution you get for y above (p), now make the substitutions a=y^2,\; b=2xy,\; c=x^2,\; k_1=f(x,y),\; k_2=g(x,y). I bet a dollar this solution will then solve the PDE for a=1.

    Yea, I know, it's not the one with variable a.. . . a journey begins with a first step.
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  3. #3
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    Alright, I'd like to know why this wouldn't work: I start with

    y^2\frac{\partial^2 p}{\partial z^2}-(z+2xy)\frac{\partial p}{\partial z}+x^2p=0

    but first solve:

    a y''-(t+b)y'+cy=0

    via power series. My initial results show the solution to be:

    y(t)=\sum_{n=0}^{\infty}k_n t^n

    where k_n=\frac{\frac{b}{a}(n-1) k_{n-1}-\frac{1}{a}k_{n-2}(c-n+2)}{n(n-1)}

    with k_0, k_1 arbitrary. I now make the back-substitutions from the PDE and obtain a power series solution in z:

    p(x,y,z)=\sum_{n=0}^{\infty}f_n(x,y)z^n

    where:

    f_n(x,y)=\frac{\frac{2x}{y}(n-1)f_{n-1}(x,y)-\frac{1}{y^2}(x^2-1)f_{n-2}(x,y)(3-n)}{y^2 n(n-1)}

    with f_0(x,y) and  f_1(x,y) arbitrary functions of x and y.
    Last edited by shawsend; November 21st 2008 at 09:35 AM. Reason: syntax errors, corrected expression for k_n
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