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Math Help - Solve the following ODE .. #5

  1. #1
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    Solve the following ODE .. #5

    Problem:
    Solve the following equation:
    \dfrac{dy}{dx}+2 \, \dfrac{sin(x)}{x} \, y = x y^2

    Solution:
    Its Bernoulli's Equation in y

    Let t=y^{-1} \implies \dfrac{dy}{dx}=\dfrac{-1}{t^2} \, \dfrac{dt}{dx}

    Multiply the equation by y^{-2} :

    y^{-2} \, \dfrac{dy}{dx}+2 \, \dfrac{sin(x)}{x} \, y^{-1} = x

    Which is:

    -t^2 \, \dfrac{1}{t^2} \, \dfrac{dt}{dx} + 2 \, \dfrac{sin(x)}{x} \, t = x

    Or:

    \dfrac{dt}{dx} - 2 \dfrac{sin(x)}{x} \, t = -x

    Which is a Linear Equation in t, the problem here is I can't find the Integrating Factor

    Since I can't find \int \dfrac{sin(x)}{x} \, dx

    Maybe I did something wrong?
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  2. #2
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    What you have found so far is fine.

    I suggest writing \displaystyle{\frac{\sin{x}}{x}} as a Taylor series, then you can integrate it term by term...
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  3. #3
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    Thanks.
    I said there is something wrong.
    I will check my TA, since he said in this course we will solve ,through the ODEs, only elementary integrals.
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