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Math Help - Using seperation of variables?

  1. #1
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    Using seperation of variables?

    What is the procedure or what are the steps to be carried out here:
    \frac{X\prime}{X}=\frac{1}{4}(\frac{Y}{Y\prime})=\  lambda
    The answer is given as X=e^{\lambda\alpha} and Y=e^{\frac{\beta}{4\lambda}}
    I don't know how to get to the answer, i have done seperation of variables, but not when i have a derivative as a denominator!
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  2. #2
    Senior Member AllanCuz's Avatar
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    Quote Originally Posted by punkstart View Post
    What is the procedure or what are the steps to be carried out here:
    \frac{X\prime}{X}=\frac{1}{4}(\frac{Y}{Y\prime})=\  lambda
    The answer is given as X=e^{\lambda\alpha} and Y=e^{\frac{\beta}{4\lambda}}
    I don't know how to get to the answer, i have done seperation of variables, but not when i have a derivative as a denominator!
    This produces 2 equations

     X^{ \prime } - \lambda X = 0

    and

    \lambda Y^{ \prime } - \frac{Y}{4} = 0

    This can become

     Y^ { \prime } - \frac{ Y }{ \lambda 4} = 0

    To solve both,

    let  X=e^{rx}

    let  Y=e^{qy}

    Therefore the first equation becomes

    re^{rx} - \lambda e^{rx} = 0

    r= \lambda

    Hence,

    X=C_1 e^{ \lambda x }

    Of course the co-efficient is just a constant to represent the general solution. Which can be re-written in the form that they gave you.

    The same follows for the Y equation!
    Last edited by AllanCuz; April 19th 2010 at 04:17 PM.
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