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Math Help - a partial differentiation problem

  1. #1
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    Question a partial differentiation problem

    Hi guys, i ve got a problem and i've never seen this type of question before..could u help me out?

    thank you
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  2. #2
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    That is in the form of a solution to a first order PDE with constant coefficients:

    a u_x+bu_y+cu=f(x,y)

    in which you let w=bx-ay and  z=y and v(w,z)=u(x,y) then express the PDE in u in terms of v. If I then solve the equation:

    2u_x+3u_y=0

    using this method, I get 3v_z=0 or v(w,z)=C(w) where C(w) is an arbitrary function of w=3x-2y that is, the function u(x,y)=C(3x-2y) solves my PDE like u(x,y)=3x-2y, u(x,y)=\sin(3x-2y), etc.

    My favorite Basic PDE book is "Basic Partial Differential Equations" by Bleecker and Csordas which goes over this subject nicely I think. It's an easy read and I recommend it if you're studying the subject.
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  3. #3
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    thanks for the reply, but i dont quite understand how to express pde in terms of v?
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  4. #4
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    Quote Originally Posted by lollol View Post
    thanks for the reply, but i dont quite understand how to express pde in terms of v?
    Let's try another way. If u(x,y) = \phi(3x-2y) then taking an x and y derivative gives

    u_x = 3\phi'(3x-2y) and u_y = -2\phi'(3x-2y).

    Now, what linear combination

    a u_x + b u_y = 0 identically (i.e. no \phi')?
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