The general second-order Fuchsian equation with regular singularities at , , and at , is given by
The exponents at the finite singularities are and those at are , where
The three sets of parameters comprise the singularity parameters , the exponent parameters , and the free accessory parameters . With and the total number of free parameters is . Heun’s equation (31.2.1) corresponds to .
An algorithm given in Kovacic (1986) determines if a given (not necessarily Fuchsian) second-order homogeneous linear differential equation with rational coefficients has solutions expressible in finite terms (Liouvillean solutions). The algorithm returns a list of solutions if they exist.
For applications of Kovacic’s algorithm in spatio-temporal dynamics see Rod and Sleeman (1995).