(Solved): \[ 1 x^{2} y^{\prime \prime}+1 x y^{\prime}+\left(x^{2}-9\right) y=0 \] Use an initial index of \( ...

\[ 1 x^{2} y^{\prime \prime}+1 x y^{\prime}+\left(x^{2}-9\right) y=0 \] Use an initial index of \( k=2 \) to develop the recurrence relation. The fillicial roots are(in ascending order) \( r_{1}=r_{2}= \) Corresponding to the larger indicial root, the recurrence relation of the solution is given by \( c_{k}=c_{k-2} \). The initial index is \( k= \) The solution is \[ y_{1}=c_{0}\left(\alpha_{1} x^{\beta_{1}}+\alpha_{2} x^{\beta_{2}}+\alpha_{3} x^{\beta_{3}}+\alpha_{4} x^{\beta_{4}}+\alpha_{5} x^{\beta_{5}}+\ldots\right) \] where \( \alpha_{1}=, \beta_{1}=, \alpha_{2}=, \beta_{2}=, \alpha_{3}=, \beta_{3}=, \alpha_{4}=, \beta_{4}=, \alpha_{5}= \), and \( \beta_{5}= \) Note: the \( \alpha_{i} \) 's and \( \beta_{i} \) 's are constants.