1 Frobenius series. Indicial equation with repeated root

1.1 Example 1. homogeneous ode \(x^{2}y^{\prime \prime }\left ( x\right ) +xy^{\prime }\left ( x\right ) +x^{2}y\left ( x\right ) =0\)
1.2 Example 2. inhomogeneous ode example \(x^{2}y^{\prime \prime }\left ( x\right ) +xy^{\prime }\left ( x\right ) +x^{2}y\left ( x\right ) =\sin \left ( x\right ) \)
1.3 Example 3. homogeneous ode example \(\left ( e^{x}-1\right ) y^{\prime \prime }\left ( x\right ) +e^{x}y^{\prime }\left ( x\right ) +y\left ( x\right ) =0\)