2 Frobenius series where indicial equation roots differ by an integer

2.1 Example 1. homogeneous ode where log term is needed \(x^{2}y^{\prime \prime }\left ( x\right ) -xy\left ( x\right ) =0\)
2.2 Example 2. homogeneous ode where log term is needed \(x^{\frac {3}{2}}y^{\prime \prime }+y=0\)
2.3 Example 3. homogeneous ode where log term is not needed \(x^{2}y^{\prime \prime }+3xy^{\prime }+4x^{4}y=0\)
2.4 Example 4 inhomogeneous ode example where log term is needed \(xy^{\prime \prime }+y=x\)