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Homework Help: Ln(y) and exp(y) :S

  1. Nov 30, 2009 #1
    Ive got the equation: sin(x) = ln(y) + (y^2)/2 + k

    The k is a constant from an earlier integration. How do I isolate y? What makes it hard for me is that if i want to get rid of ln() i need to use exp() but then the other y is in exp() and if i want to get rid of that, the first y is in ln() again.

    Sorry, for the bad english =) I hope you can help.
  2. jcsd
  3. Nov 30, 2009 #2


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    Homework Helper

    I'm afraid it's not possible to isolate y algebraically.

    Taking the exponential of both sides: [tex]y=e^{sinx-\frac{y^2}{2}-k}=\frac{e^{sinx-k}}{e^{\frac{y^2}{2}}}[/tex]

    So now you have: [tex]ye^{\frac{y^2}{2}}=e^{sinx-k}[/tex]

    There is some method to solving for x in: [itex]xe^x=y[/itex]
    but the name of it has slipped my mind and whether it can be adapted to solve for y in your problem I'm unsure of as well. Hopefully someone else can help you with this.
  4. Nov 30, 2009 #3


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    Science Advisor

    Generally speaking, when you have a variable both "inside" and "outside" a transcendental function, there is no algebraic way to isolate that variable.

    The "method of solving [itex]xe^x= y[/itex]" is "Lambert's W function" which is defined as the inverse function to [itex]xe^x[/itex]. That is, [itex]W(xe^x)= x[/itex] so [itex]W(xe^x)= W(y)[/itex] and [itex]x= W(y)[/itex].
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