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Playing With Logs while solving y-x=5(ln|y+3|-ln|x+4|) for y.

  1. Sep 25, 2013 #1
    1. The problem statement, all variables and given/known data

    The problem is actually the following Separable Differential Equation:
    [itex]\frac{dy}{dx}=\frac{xy+3x-y-3}{xy-2x+4y-8}[/itex]​
    I am required to find y(x).


    2. Relevant equations and techniques

    - factorization (applied on the numerator and the denominator in the problem equation)
    - basic integration (applied after separating the variables)
    - at least the following logarithmic identities:
    [itex]e^{ln|x|}=x[/itex] ; [itex]ln|xy|=ln|x|+ln|y|[/itex]​


    3. The attempt at a solution

    [itex]\frac{dy}{dx}=\frac{xy+3x-y-3}{xy-2x+4y-8}[/itex]

    [itex](1-\frac{5}{y+3})dy=(1+\frac{5}{x+4})dy[/itex]

    [itex]y-5ln|y+3|=x+5ln|x+4|+C[/itex]

    [itex]y-x=5(ln|y+3|-ln|x+4|)[/itex]


    I also put this result in Wolfram Alpha, but it could not solve the equation for y. My instructor apparently believes that an y=f(x) can be obtained. Is there any way to do so?
     
    Last edited: Sep 25, 2013
  2. jcsd
  3. Sep 25, 2013 #2

    haruspex

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    You appear to have a sign wrong in the middle two equations, but since it comes out right at the end maybe this was a copying out error.
    i don't see a way to get it into y = f(x) form either.
     
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