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Seperable Equations using a substitution for a differential equation

  1. Jul 10, 2008 #1
    1. The problem statement, all variables and given/known data

    The d.e

    y' = (y+2x)/(y-2x)

    is NOT seperable, but if you use a substitution then you obtain a new d.e involving x and u, then the new d.e is seperable.... Solve the original d.e by using this change of variable method

    2. Relevant equations
    I'm going to use the substitution that u=y/x in the form y=ux

    3. The attempt at a solution

    y' = (y+2x)/(y-2x)
    let y=ux

    y' = (ux+2x)/(ux-2x)
    y'(ux-2x) - (ux+2x) = 0 <--- thus it is seperable
    So I can say

    d/dx((ux^2)/2 - x^2) - d/dx((ux^2/2) + x^2) = 0

    Putting it all together

    d/dx[(ux^2)/2 - x^2 - (ux^2)/2 - x^2] = 0
    d/dx (-2x^2) = 0

    Well that is where I am stuck, how do I solve it from there and what am I trying to get because I've changed the d.e so I'm not sure how the answer from the new d.e will help me find a solution to the original one?
  2. jcsd
  3. Jul 10, 2008 #2


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    Homework Helper

    If you are going to substitute y=ux then you also need to substitute y'=(ux)'=u'x+u. That will change the differential equation for y into one for u. You shouldn't have a y' hanging around after the substitution.
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