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DE question

  1. Nov 24, 2008 #1
    1. The problem statement, all variables and given/known data
    Solve: [​IMG]

    2. Relevant equations
    DE stuff

    3. The attempt at a solution
    I started by doing substitution: x=U+3/2 and y=W-1/2

    So, it gave me [​IMG], but [​IMG], so [​IMG].

    Therefore, [​IMG]

    And, [​IMG]. Substitution z=W/U. Giving, [​IMG] (*).

    And, [​IMG]

    Rearranging gives [​IMG]. Substitute in (*)

    [​IMG] => [​IMG] => [​IMG]

    Dividing 1/(1+z2) and -z/(1+z2) and integrating gives:

    [​IMG]. But now I'm stuck :( because I can't solve this for z and go back to y and x :( maybe I made an arithmetical mistake or maybe there is a better method?
    Thank you
    Last edited: Nov 24, 2008
  2. jcsd
  3. Nov 24, 2008 #2
    I don't know how correct the method you detailed is but a lot of times for a differential equation, an implicit solution is sufficient.
  4. Nov 25, 2008 #3
    Surely there is a way to solve it explicitely
  5. Nov 25, 2008 #4


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    Staff Emeritus
    Science Advisor

    Why "surely". You don't need to solve
    [tex]arctan z-\frac{1}{2}ln(z^2+ 1)= ln(U)+ C[/tex]

    z= W/U so that is
    [tex]arctan W/U-\frac{1}{2}ln(\frac{W^2}{U^2}+ 1)= ln(U)+ C[/tex]
    and U= x- 3/2, W= y+ 1/2 so
    [tex]arctan\frac{y+ 1/2}{x-3/2}- \frac{1}{2}ln(\frac{(y+1/2)^2}{(x-3/2)^2}= ln(x- 3/2)+ C[/tex]
    That's a perfectly good solution. (Assuming your integration was correct.)
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