Maple and Mathematica both did quite well in solving differential equations with less than one percentage point difference in solving the almost 16,000 problems collected from different textbooks.
The solution of each ODE was simplified. Maple’s leaf size of its result is about 66% smaller than Mathematica’s. Maple was faster in the overall time it took to complete all the tests. All tests were run on same PC with 128 GB of RAM. These tests only check that a solution was obtained for a differential equation within the timelimit. No grading on quality of solution is made and no verification is made that the solution is correct or not.
Only the time used for solving the ODE was taken into account. The time used for the simplification of the solution is not counted.
One of the areas that Maple lacked Mathematica the most was in solving an ODE using series option. Maple series option does not support solving an ODE where the expansion point is irregular singular point (also called an essential singularity).
These problems can not be solved using power series or Frobenius series methods but require asymptotic series methods which currently not supported by Maple’s dsolve command using the series option.
Mathematica has a special dsolve command AsymptoticDSolveValue for this purpose. This
is why Maple scored about 95% while Mathematica’s scored almost 100% in this area of the
tests.
It is possible to write dedicated code in Maple to solve such problems, but not using the default dsolve command with the series option whcih is the only command used in these tests.
In the area of solving system of differential equations, the results were very close, with Mathematica solving 96.46% and Maple solving 96.75%.
For first order ODE’s that are nonlinear in \(y'(x)\), Maple edged Mathematica by little over 2% while for first order ODE’s that are linear in \(y'(x)\) Maple score was 1.5% higher than Mathematica.
For second and higher order ODE’s that are nonlinear, Maple edged Mathematica by almost 4.4%.
For second and higher order ODE’s that are linear, Maple and Mathematica score was almost the same with Maple scoring about a fraction of 1% better than Mathematica.
Overall, Maple performed better in the area of nonlinear differential equations and Mathematica performed better in the area of solving using series when the expansion point is irregular singular point.