How can I get the equations of motion used in webots?

EDIT: the below mention function does not export equations directly, as you requested, but exports the dynamic system in the DIF - Dynamics Interchange Format

Webots uses the Open Dynamics Engine (ODE) to simulate physics. ODE includes an export function for exporting the ODE called dWorldExportDIF().

Webots seems to support this export function and it also includes a test, which uses the function mentioned above to export the equations. The test, which exports the equations can be found here: https://github.com/cyberbotics/webots/blob/master/tests/physics/plugins/physics/ode_dif_exporter/ode_dif_exporter.c