New Techniques to Solve Differential Equations Automatically

Solving differential equations is a very important task in mathematics, science and engineering. Laplace transform is a well known and very widely used method for solving differential equations; although Laplace transform is very powerful and widely used in that regard, however, it suffers a serious drawback, which is the calculation of inverse Laplace transform. Such kind of inverse calculation is problematic or simply impossible, except some very basic functions. Sumudu transform is a new integral transform with new features. In this work, a new computational method is proposed to solve differential equations, the new method combines both Laplace transform and Sumudu transform such that the calculation of the inverse Laplace transform can be avoided; it is also shown that the traditional method by power series can be improved. Furthermore, it is demonstrated that the new method and techniques presented in this work can be implemented in computer algebra systems such as Maple to solve a wide range of differential equations automatically.