Symbolic Analysis of Second-order Ordinary Differential Equations with Polynomial Coefficients

The singularity structure of a second-order ordinary differential equation with polynomial coefficients often yields the type solution. It is shown that $\theta$-operator method can be used as symbolic computational approach to obtain indicial and recurrence relation. Consequently, leads transformations yield solution in terms special function, if suitable. Hypergeometric Heun-type equations are mostly employed physical applications. Thus, only these their confluent types considered SageMath routines which assembled open-source package symODE2.