Differential Equations

Maple 10 uses new sophisticated functions to represent solutions to many formerly unsolvable families of linear and nonlinear differential equations. Original computational algorithms have been developed by the Maplesoft research team–the most important development in ordinary differential equations (ODEs) in the last four Maple releases.

The following is a list of some of the improvements for finding exact solutions of ODEs and partial differential equations (PDEs), and systems of ODEs and PDEs.

• Newly developed solving algorithms for certain classes of Abel type ODEs, Riccati type ODEs, and Heun function solutions
• Newly developed solving algorithms for computing different types of Traveling Wave Solutions for PDEs and PDE systems
• Newly added algorithms for computing Liouvillian solutions and for solving a general class of ODEs that can be mapped to Abel type
• Improved algorithms for computing doubly periodic solutions for second order linear ODEs

Maple 10 also includes improvements and additions to the DEtools and PDEtools packages.

• Nine new commands added to DEtools:
  • Compute homomorphisms between the solution spaces of two linear differential operators
  • Compute particular solutions to linear equations in cases where the general solution cannot be found
  • Handle differential rational normal forms and hyperexponential functions
• New differential polynomial extensions added to PDEtools for differential polynomial forms. The interactive ODE Analyzer takes full advantage of the new algorithms.