2 Frobenius series where indicial equation roots differ by an integer

2.1 Example 1. homogeneous ode where log term is needed x2y(x)xy(x)=0
2.2 Example 2. homogeneous ode where log term is needed x32y+y=0
2.3 Example 3. homogeneous ode where log term is not needed x2y+3xy+4x4y=0
2.4 Example 4 inhomogeneous ode example where log term is needed xy+y=x