Originally Posted by

**garunas** Let $\displaystyle \rho \in \mathbb{R} $ and let

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

be a solution to the differential equation:

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

The first question asks: "What is the set of possible values for $\displaystyle \rho$?" For which i have found that

$\displaystyle \rho = -\sqrt{3}$ and $\displaystyle \rho = \sqrt{3}$

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

$\displaystyle 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?