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Homework Help: Cauchy Problem

  1. Aug 26, 2010 #1
    1. The problem statement, all variables and given/known data


    getting gen sol of

    xy3zx+x2z2zy=y3z

    solve cauchy problem
    x=y=t, z=1/t


    3. The attempt at a solution

    i got gen sol F(C1,C2)=0 as

    C1=x/z, C2=y4-x2z2

    i inserted t for x and y and 1/t for z and ended up with

    C1-2=1/(C22)

    I'm unsure what to do from here i tried to get equation in terms of z by inserting eqns for C1 and C2 but it becomes very messy
     
  2. jcsd
  3. Aug 26, 2010 #2
    Try to learn to not let messy intimidate you. Just muscle through it if you have to but you don't have to in this case. Just leave it in it's implicit form:

    [tex]F(C1,C2)=F(x/y,y^4-x^2 z^2)=0[/tex]

    and often you won't be able to solve explicitly for z(x,y) anyway.

    Here's a back-substitution check of [itex]C1+C2^2=0[/itex] in Mathematica. Note how [itex]y^3 z[/itex] is obtained after evaluating the left side as expected.

    try and follow it and change the expression for F(C1,C2) if you wish:

    Code (Text):
    In[40]:=
    c1 = x/z[x, y];
    c2 = y^4 - x^2*z[x, y]^2;
    myexp = c1 + c2^2 == 0;
    myzx = First[D[z[x, y], x] /.
        Solve[D[myexp, x], D[z[x, y], x]]]
    myzy = First[D[z[x, y], y] /.
        Solve[D[myexp, y], D[z[x, y], y]]]
    Simplify[x*y^3*myzx + x^2*z[x, y]^2*myzy]

    Out[43]=
    -((z[x, y]*(1 - 4*x*y^4*z[x, y]^3 +
         4*x^3*z[x, y]^5))/
       (x*(-1 - 4*x*y^4*z[x, y]^3 +
         4*x^3*z[x, y]^5)))

    Out[44]=
    -((8*y^3*z[x, y]^2*(y^4 - x^2*z[x, y]^2))/
       (x*(-1 - 4*x*y^4*z[x, y]^3 +
         4*x^3*z[x, y]^5)))

    Out[45]=
    y^3*z[x, y]

     
     
    Last edited: Aug 26, 2010
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