Differential–anti-differential equations and their solutions

Abstract

In this work, we extend the theory of differential equationsto differential–anti-differential equations and solve several types of such equations. To do this, first, we define a derivative– antiderivative operator with a base function and express derivatives and antiderivatives in the form of this operator. Next, we come to differential–anti-differential equations. To solve some of these equations, first, we investigate the Auxiliary equation(s) and find the roots. Roots are then used in the base function to get the exact solutions of the differential– anti-differential equation. The process can be used to solve several well-known differential equations such as Continuity, Heat, Wave, Laplace, Schrodinger, Euler, Blasius, etc. Our technique shows that every elementary function can solve several types of linear and nonlinear ordinary as well as partial differential–anti-differential equations