Detailed description of the Kovacic algorithm with worked out examples were given. All three cases of the Kovacic algorithm were implemented using object oriented design in Maple. The software was then used to analyze over \(3000\) differential equations. The results showed that case one and two combined provided coverage for \(99.9\)% of the ode’s with \(97.36\)% of the ode’s solved using case one algorithm and \(2.54\)% solved using case \(2\) algorithm with only \(0.1\)% requiring case \(3\). Not a single ode was found that required the use of case three with \(n=6\) or \(n=12\).
One restriction found on the use of the algorithm is that it requires an ode with its coefficients being numerical and not symbolic. This is because the algorithm has to decide in step \(2\) if \(d\) (the degree of polynomial \(p(x)\)) is non-negative integer or not in order to continue to step \(3\). If some of the ode coefficients were symbolic, it will not be able to decide on this (without additional assumptions provided). Therefore this algorithm works best with ode’s having its coefficients given with numerical values.