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Math Help - partial differential equations

  1. #1
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    partial differential equations

    find general solution of:

    u_x + x u_y + xy u_z = xyzu

    i've tried with this

    dx=dy/(x)=dz/(xy)=du/(xyzu) ...but didn't get the correct solution

    i appreciate any help...thanks
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  2. #2
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    Quote Originally Posted by tom007 View Post
    find general solution of:

    u_x + x u_y + xy u_z = xyzu

    i've tried with this

    dx=dy/(x)=dz/(xy)=du/(xyzu) ...but didn't get the correct solution

    i appreciate any help...thanks
    Your method is correct! I got

    u = e^{z^2/2}F(x^2-2y,y^2-2z).

    What did you get?
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  3. #3
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    i don't know how to solve this..i thought F should have 3 components.. i got many different results...
    could u pls write the steps, if it's not the problem..
    thank you very much Danny!
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  4. #4
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    Quote Originally Posted by tom007 View Post
    i don't know how to solve this..i thought F should have 3 components.. i got many different results...
    could u pls write the steps, if it's not the problem..
    thank you very much Danny!
    Picking it up from your first post - the characteristic equations are:

     <br />
\frac{dx}{1} = \frac{dy}{x} = \frac{dz}{xy} = \frac{du}{xyz} <br />

    We'll pick in pairs

    1)  <br />
\frac{dx}{1} = \frac{dy}{x} \;\; \text{so}\;\; x^2 - 2y = c_1<br />

    2)  <br />
\frac{dy}{x} = \frac{dz}{xy} \;\; \text{so}\;\;y^2 - 2z = c_2

    3)  <br />
\frac{dz}{xy} = \frac{du}{xyzu}\;\; \text{so}\;\; ue^{-z^2/2} = c_3<br />

    The solution c_3 = F(c_1,c_2) gives ue^{-z^2/2} = F(x^2 - 2y , y^2 - 2z) leading to my solution.
    Last edited by Jester; January 13th 2010 at 02:36 PM.
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  5. #5
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    wow..that's so simple, i was complicating ..better not to say!!
    thanks Danny!!

    do u maybe know other 2 exercise i've posted?
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  6. #6
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    ey Danny...how could i check this result?

    is there any formula?
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  7. #7
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    Quote Originally Posted by tom007 View Post
    ey Danny...how could i check this result?

    is there any formula?
    The easiest way to check is to substitute the solution into the PDE itself and check that it's satisfied.
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  8. #8
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    i know that...but how should i substitute F function?
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