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Partial diffs help please!

  1. Aug 27, 2011 #1
    Show that under the transformation

    u=x, v=[itex]\alpha[/itex]x+[itex]\beta[/itex]t​
    Ayxx+Byxt+Cytt=0 ; B^2-4AC>0
    becomes

    Ayuu+(2A[itex]\alpha[/itex]+B[itex]\beta[/itex])yuv+(A[itex]\alpha[/itex]2+B[itex]\alpha[/itex][itex]\beta[/itex]+C[itex]\beta[/itex]2)yvv=0

    (A,B,C are constants)
    I have no idea where to start. and i have to present this problem to the front of my class on monday. Can anyone give me a big head start or anything?
     
  2. jcsd
  3. Aug 27, 2011 #2
    gotta' start taking partials. That's where to start.

    Is it like a bunch of people in there?

    Ok, just playing.

    So if y=f(x,t) and x=u and v=ax+bt, then:

    [tex]y_x=y_u u_x+y_v v_x[/tex]
    [tex]y_x=y_u+a y_v[/tex]

    but the second one is a litle tricky since you taking the partial of partials so:

    [tex]y_{xx}=\frac{\partial}{\partial x} \left(y_u+a y_v\right)=y_{uu} u_x+y_{vu} v_x+a\left(y_{uv}u_x+y_{vv} v_x\right)[/tex]

    Ok then, keep doing that for each partial in the first expression, make all those substitutions back into the first expression which wil turn it into an expression of y in terms of u and v, then simplify, cancel, whatever, and it should look like the second expression.
     
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