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Separable equations proving

  1. Oct 3, 2011 #1
    1. The problem statement, all variables and given/known data
    If y(1+x2) dy/dx = 2x (1-y2), prove that (1+x2)2(1-y2)=A, where A is constant.

    2. Relevant equations
    Separable equations

    3. The attempt at a solution

    Separate the terms:

    y/(1-y2) dy = 2x/(1+x2) dx

    Integrating both sides will get:

    ∫ y/(1-y2) dy = ∫ 2x/(1+x2) dx

    Use substitution method for ∫ y/(1-y2) dy:

    u = 1-y2
    du = -2y dy
    -du/2 = y dy

    ∫ -u/2 du = -1/2 ∫ u du
    = (-1/2)*(u2/2)
    = -u2/4 + C
    = -(1-y2)2/4

    Use substitution method for ∫ 2x/(1+x2) dx:

    u= 1+x2
    du = 2x

    ∫ 1/u du = ln u + C
    = ln (1+x2)

    Putting them back together will get:

    -(1-y2)2/4 = ln (1+x2)

    I'm pretty much unable to continue from here.
  2. jcsd
  3. Oct 3, 2011 #2


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    Science Advisor
    Homework Helper

    Don't you mean 1/u there as well?
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