6.2.4 Statistics and discussion of results obtained using Kovacic algorithm
This gives summary of results obtained using testsuite of \(3000\) differential equations, all of which were selected as linear with rational coefficients as functions of \(x\) that can be solved using this algorithm.
The ode’s used in the testsuite were collected by the author and stored in sql database. These were collected from a number of standard textbooks and other references such as “Differential Equations. E. Kamke. 3th edition. Chelsea.” and “Ordinary Differential Equations And Their Solutions. Murphy, George Moseley. Dover. 2011”.
All the ode’s were successfully solved using the Kovacic algorithm as implemented here and each solution was verified using Maple odetest.
The following diagram shows the percentage of ode’s solved using each case.
Case \(3\) was required for solving only \(3\) odes. It used \(n=4\) for all \(3\) ode’s. \(n=6\) and \(n=12\) were not reached or required to try. Recall that \(n\) for case \(3\) is the degree of the polynomial in \(\omega \) used to solve for in order to find the \(z\) solution from \(z=e^{\int \omega \,dx}\).
This result shows that case \(1\) and \(2\) combined is all what is needed to solve \(99.9\)% of ode’s used in practice. Larger collection of ode’s than the \(3000\) used could produce different results, but the overall trend is that case \(3\) is rarely needed in practice and within case \(3\), \(n=6\) and \(n=12\) are even less likely to be required.
When forcing the algorithm to use case \(3\) and only use \(n=12\), this resulted in a very long computation time on some ode’s. For an example, using ode \(y''+xy'+y=0\) which satisfies all three cases, and asking the solver to use case \(3\) and \(n=12\), it was found that it required \(p(x)\) of degree \(d=24\) in order to find \(\omega \) of degree \(12\) that can be solved. The total number of trials in step 3 of case three to find such solution was found to be \(2367\). This took over 30 minutes to complete.
In comparison, the same ode was solved using case one in less than one second giving the same solution on the same computer.
The testsuite also calculates the distribution of cases which has its necessary conditions satisfied for each ode. Recall that having the necessary conditions for a case satisfied does not mean a solution would be found using that case. The following bar chart shows the percentages of the \(3000\) ode’s that satisfied the necessary conditions each case. This chart shows that many ode’s satisfy the conditions for more than one case at the same time.
