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Math Help - solving semilinear PDE

  1. #1
    Member Jason Bourne's Avatar
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    solving semilinear PDE

    Using the method of characteristics, show that the semilinear PDE

    (x+y)u_x + (x-y)u_y = 2x

    has the general solution H(u-x-y, y^2 + 2xy -x^2) = 0<br />

    Find the solution for the case u(x,0) = 2x
    Last edited by Jason Bourne; November 27th 2008 at 06:03 AM.
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  2. #2
    Member Jason Bourne's Avatar
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    I think I need to solve the differential equation

    \frac{dy}{dx} = \frac{x-y}{x+y} <br />

    Does anyone know how to solve the above equation to get y as a function of x?
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  3. #3
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    Quote Originally Posted by Jason Bourne View Post
    I think I need to solve the differential equation

    \frac{dy}{dx} = \frac{x-y}{x+y} <br />

    Does anyone know how to solve the above equation to get y as a function of x?
    It's homogeneous: (x+y)dy=(x-y)dx so let y=vx. Gets a little messy but I get y^2+2xy-x^2=h. Still though I think we could let w=y^2+2xy-x^2 and z=y to solve the PDE. I'm probably not doing it the way you want but I get for one solution:

    u(x,y)=2\left(y+1/2\sqrt{y^2-2xy+x^2}\right)+f(y^2+2xy-y^2) and supplying your side condition, then f(x,y)=0
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