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Math Help - partial differential equation

  1. #1
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    partial differential equation

    Hi, I have the differential equation
     \frac{ \partial z }{ \partial x } + \frac{ \partial z }{ \partial y } = z^2 . Satisfied by  z(x,y) , where  z = f(x) on y=0 and x is between minus to plus infnity, show that the solution is
     z = \frac{f(s)}{1-f(s) t} , where x=t+s and y=t.

    I know I have to use the chain rule but I am not sure how to start. Do you have any hints you could give me to start working the solution? Thanks a lot.
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  2. #2
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    Quote Originally Posted by mlazos View Post
    Hi, I have the differential equation
     \frac{ \partial z }{ \partial x } + \frac{ \partial z }{ \partial y } = z^2 . Satisfied by  z(x,y) , where  z = f(x) on y=0 and x is between minus to plus infnity, show that the solution is
     z = \frac{f(s)}{1-f(s) t} , where x=t+s and y=t.

    I know I have to use the chain rule but I am not sure how to start. Do you have any hints you could give me to start working the solution? Thanks a lot.
    Probably the simplest approach would be to solve x = t + s, y = t for t and s in terms of x and y. Then plug that into your solution and you've got z(x, y). Taking the partials should be easy now.

    -Dan
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    So we just need to verify that it is a solution or we maybe there is a way to find the solution using a method or something? If I show that the solution satisfies the partial differential equation is it sufficient? Thanks again.
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  4. #4
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by mlazos View Post
    So we just need to verify that it is a solution or we maybe there is a way to find the solution using a method or something? If I show that the solution satisfies the partial differential equation is it sufficient? Thanks again.
    My bad. I thought all you had to do was verify. Your equation is a first order quasilinear equation. One way to solve it is to use the method of characteristics which from the form of the solution was used here.

    -Dan
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