91Ó°ÊÓ

The power series solution for the given ODE \(\left(x^{2}+1\right) y^{\prime \prime}+x y^{\prime}+x y=0\) can be found by assuming the solution in the form of a power series \(y(x) = \sum_{n=0}^{\infty} a_n x^n\) and calculating its first and second derivatives. Substitute the power series and their derivatives into the given ODE and combine the sums. The coefficients of the power series can be determined iteratively using the relation \(a_{n+2} = -\frac{(n+1)a_{n+1} + a_{n+1}}{n(n-1)}\). The final solution will have the form \(y(x) = a_0 y_1(x) + a_1 y_2(x)\), where \(y_1(x)\) and \(y_2(x)\) are linearly independent power series solutions constructed using the coefficients found from the relation provided.