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Help Solving this PDE

  1. Feb 16, 2016 #1
    1. The problem statement, all variables and given/known data
    Solve y*∂Ψ/∂x-(x/3)∂Ψ/∂y

    2. Relevant equations


    3. The attempt at a solution
    My teacher told me to try separation of variables but and I tried to set Ψ=X(x)Y(y) where X is a function of just X and Y is a function of just y but when I got the solution and put it into the original pde it did not work.
     
  2. jcsd
  3. Feb 16, 2016 #2

    blue_leaf77

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    Where is the equal sign?
    Can you post your work as well as the answer you got?
     
  4. Feb 16, 2016 #3
    Sorry the equal sign might help The equation is
    y*∂Ψ/∂x-(x/3)*∂ψ/∂x=0



    So first I defined Ψ(x,y)=X(x)Y(y)
    thus the equation becomes
    y*∂(X(x)Y(y)/∂x-(x/3)*∂(X(x)Y(y)/∂y=0
    Rearranging and using the multiplication rule
    y*Y(y)d(X(x))/dx=(x/3)*X(x)d(Y(y))/dy
    Rearranging again
    y*(1/X(x))d(X(x))dy=(x/3)*(1/Y(y))*d(Y(y))dx
    then integrating
    y^2/2*ln(X(x))=x^2/6*ln(Y(y))+c

    That is as far as I got.
     
  5. Feb 16, 2016 #4

    SammyS

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    Rearrange the above so one side of the equation only depends on x and the other only on y .

    Then each side must equal a constant, Right?
     
    Last edited: Feb 16, 2016
  6. Feb 17, 2016 #5

    HallsofIvy

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    That last step is incorrect- you cannot Integrate that way!
    Instead, at the point where you have yX'/X= (x/3)Y'/Y, divide both sides by xy/3 to get 3X'/(xX)= Y'/(yY).
    The left side depends only on x while the right side depends only on y. But the equation has to be true for all x and y. Imagine changing x while not changing y. Since y does not change the right side does not change. But that means the left side cannot change! That is 3X'/(xX)= C, a constant. Since 3X'/(xX)= Y'/(yY), we also have Y'/yY= C.

    3X'/(xX)= C is the same as 3dX/dx= CxX, a separable differential equation.

    Mod note: Removed some text as being too much help.
     
    Last edited by a moderator: Feb 17, 2016
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