3 Frobenius series. Indicial equation with root that differ by non integer

3.1 Example 1. homogeneous ode \(2x^{2}y^{\prime \prime }\left ( x\right ) -xy^{\prime }\left ( x\right ) +\left ( 1-x^{2}\right ) y\left ( x\right ) =0\)
3.2 Example 2. homogeneous ode \(2x^{2}y^{\prime \prime }\left ( x\right ) -xy^{\prime }\left ( x\right ) +\left ( 1-x^{2}\right ) y\left ( x\right ) =1+x\)