1 Frobenius series. Indicial equation with repeated root

1.1 Example 1. homogeneous ode x2y(x)+xy(x)+x2y(x)=0
1.2 Example 2. inhomogeneous ode example x2y(x)+xy(x)+x2y(x)=sin(x)
1.3 Example 3. homogeneous ode example (ex1)y(x)+exy(x)+y(x)=0