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Thread: solving differential equation

  1. #1
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    solving differential equation

    1. Find the general solution of the following differential equations:
    (a) dy/dx = y(1-y)
    (b) dy/dx = 3 -2y

    what i tried to do was move them to LHS....

    [$\displaystyle \frac{1}{y(1-y)}$]($\displaystyle \frac{dy}{dx}$)=1

    Integrate wrt x,

    $\displaystyle \int$$\displaystyle \frac{1}{y(1-y)}$ dy=$\displaystyle \int$1 dx

    then...i'm stuck...

    as for (b)
    2y + dy/dx =3
    ....I couldn't figure it out how to express y in terms of x eventually.


    any help is much appreciated.
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  2. #2
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    The RHS is easy, $\displaystyle \displaystyle \int{1\,dx} = x + C_1$.

    The LHS is solved using Partial Fractions:

    $\displaystyle \displaystyle \frac{A}{y} + \frac{B}{1-y} = \frac{1}{y(1 - y)}$

    $\displaystyle \displaystyle \frac{A(1-y)+By}{y(1-y)} = \frac{1}{y(1-y)}$

    $\displaystyle \displaystyle A(1-y)+By=1$

    $\displaystyle \displaystyle A-Ay+By =1$

    $\displaystyle \displaystyle (B-A)y + A = 0y + 1$

    $\displaystyle \displaystyle B-A=0$ and $\displaystyle \displaystyle A=1$.

    So $\displaystyle \displaystyle A=1, B=1$.


    Therefore the LHS:

    $\displaystyle \displaystyle \int{\frac{1}{y(1-y)}\,dy} = \int{\frac{1}{y} + \frac{1}{1-y}\,dy}$

    $\displaystyle \displaystyle = \ln{|y|} + \ln{|1-y|} + C_2$

    $\displaystyle \displaystyle =\ln{\left|y(1-y)\right|} + C_2$.



    So your DE solution is

    $\displaystyle \displaystyle \ln{|y(1-y)|} +C_2 = x+C_1$

    $\displaystyle \displaystyle \ln{|y(1-y)|} = x + C$ where $\displaystyle \displaystyle C = C_1-C_2$

    $\displaystyle \displaystyle |y(1-y)| = e^{x + C}$

    $\displaystyle \displaystyle |y(1-y)| = e^Ce^x$

    $\displaystyle \displaystyle y(1-y) = Ae^x$ where $\displaystyle \displaystyle A = \pm e^C$

    $\displaystyle \displaystyle y - y^2 = Ae^x$

    $\displaystyle \displaystyle y^2 - y = -Ae^x$

    $\displaystyle \displaystyle y^2 - y + \left(-\frac{1}{2}\right)^2 = \left(-\frac{1}{2}\right)^2 - Ae^x$

    $\displaystyle \displaystyle \left(y-\frac{1}{2}\right)^2 = \frac{1}{4} - Ae^x$

    $\displaystyle \displaystyle y-\frac{1}{2} = \pm \sqrt{\frac{1}{4} - Ae^x}$

    $\displaystyle \displaystyle y - \frac{1}{2} = \pm \sqrt{\frac{1-4Ae^x}{4}}$

    $\displaystyle \displaystyle y - \frac{1}{2} = \pm \frac{\sqrt{1-4Ae^x}}{2}$

    $\displaystyle \displaystyle y = \frac{1 \pm \sqrt{1 - 4Ae^x}}{2}$.
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  3. #3
    MHF Contributor chisigma's Avatar
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    Setting $\displaystyle u= y-\frac{1}{2}$ You obtain...

    $\displaystyle \displaystyle \int \frac{dy}{y\ (1-y)} = \int \frac{du}{\frac{1}{4} - u^{2}} = 2\ \tanh^{-1} 2 u + c$



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