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Homogenous Differential Equations (Converting Back into Explicit Form

  1. Jul 19, 2014 #1
    1. The problem statement, all variables and given/known data

    x*e^(y/x) + y dx = xdy, y(1) = 0

    2. Relevant equations

    3. The attempt at a solution

    To solve, I divide everything by x dx to put everything in terms of v.

    e^v + v = dy/dx

    dy/dx = x dv/dx + v

    e^v + v = x dv/dx + v

    e^v = x dv/dx

    e^v / dv = x/dx

    Flip both sides.

    e^-v dv = 1/x dx

    Integrate both sides

    -e^-v = ln|x| + c

    -e^(-y/x) = e^(ln|x| + c)

    -e^(-y/x) = x * e^c

    -e^(-y/x) = cx

    ln(-e^(-y/x)) = ln(cx)

    ln(e^(x/-y)) = ln(cx)

    -x/y = ln(cx)

    1/y = -ln(cx)/x

    y = -x/(ln(cx))

    Is that the correct explicit form? Would it make it easier if I used the initial condition to find c, and then attempted to put it in explicit form, or not?
  2. jcsd
  3. Jul 19, 2014 #2
    -e^-v = ln|x| + c

    -e^(-y/x) = e^(ln|x| + c)

    this step does not follow. otherwise I think you're on the right track.
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