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Math Help - Solving a differential equation using power series

  1. #1
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    Solving a differential equation using power series

    Let \rho \in \mathbb{R} and let

      y(x) = x^\rho\left(1 + \sum_{n=1}^\infty a_{n}x^n\right)\hspace{10mm}(1)

    be a solution to the differential equation:

    y''(x) + \left(1 + \frac {1}{x} \right)y'(x) - \frac{3}{x^2} y(x) = 0\hspace{10mm}(2)

    The first question asks: "What is the set of possible values for \rho?" For which i have found that
    \rho = -\sqrt{3} and  \rho = \sqrt{3}

    The second question (the one I'm stuck on) asks: "By substituting (1) into (2) for each value of \rho found in the previous question determine the corresponding coefficient a_{1} in (1). I have somewhat done this and got this equation:

    x^{\rho - 2}\left(\rho^2 - 3\right) \hspace{3mm}+\hspace{3mm} \rho x^{\rho - 1} \hspace{3mm}+\hspace{3mm}  \sum_{n=1}^\infty ((n+\rho)^2 - 3)a_{n}x^{n+\rho - 2} \hspace{3mm}+\hspace{3mm}  \sum_{n=1}^\infty \left(n + \rho)a_{n}x^{n + \rho -1} =  0

    Where do I go from here? Is my equation even right in the first place?
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  2. #2
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    Quote Originally Posted by garunas View Post
    Let \rho \in \mathbb{R} and let

      y(x) = x^\rho\left(1 + \sum_{n=1}^\infty a_{n}x^n\right)\hspace{10mm}(1)

    be a solution to the differential equation:

    y''(x) + \left(1 + \frac {1}{x} \right)y'(x) - \frac{3}{x^2} y(x) = 0\hspace{10mm}(2)

    The first question asks: "What is the set of possible values for \rho?" For which i have found that
    \rho = -\sqrt{3} and  \rho = \sqrt{3}

    The second question (the one I'm stuck on) asks: "By substituting (1) into (2) for each value of \rho found in the previous question determine the corresponding coefficient a_{1} in (1). I have somewhat done this and got this equation:

    x^{\rho - 2}\left(\rho^2 - 3\right) \hspace{3mm}+\hspace{3mm} \rho x^{\rho - 1} \hspace{3mm}+\hspace{3mm}  \sum_{n=1}^\infty ((n+\rho)^2 - 3)a_{n}x^{n+\rho - 2} \hspace{3mm}+\hspace{3mm}  \sum_{n=1}^\infty \left(n + \rho)a_{n}x^{n + \rho -1} =  0

    Where do I go from here? Is my equation even right in the first place?
    Dear garunas,

    I hope your equation is correct. By rearranging that you could obtain,

    x^{\rho-2}(\rho^2-3)+\left[\rho+\left((1+\rho)^2-3\right)a_1\right]x^{\rho-1}+\sum_{n=1}^{\infty}\left[\left((n+1+\rho)^2-3\right)a_{n+1}+(n+\rho)a_n\right]x^{n+\rho-1}=0

    Now by equating coefficients of similar powers of x to zero,

    \rho^2-3=0\rightarrow \rho=\pm\sqrt{3}

    \rho+\left((1+\rho)^2-3\right)a_1=0\Rightarrow{a_1=\frac{-\rho}{\rho^2+2\rho-2}

    When \rho=\sqrt{3},

    a_1=\frac{-\sqrt{3}}{2\sqrt{3}+1}

    When \rho=-\sqrt{3},

    a_1=\frac{\sqrt{3}}{1-2\sqrt{3}}

    Hope you understood.
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