SOLUTION OF THE SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS USING RUNGA - KUTTA METHOD BY A NEW MATHEMATICAL TECHNIQUE

This study aims to recognizing the role played by differential and normal equations loop and their methods of solution if they are linear or non-linear, their class, and if they are of constant or variable coefficient .The study also aims to study the concept and development implementation of differential equations and their increasing importance in all scientific fields and their some applications. We followed in this study the mathematical deductive and inductive by using Runga - Kutta method by a new mathematical technique . We found the some following results:.. Runga - Kutta's method in general situation depends upon calculating inclination at the point  and at many other points nearing  then taking the average of these inclinations and multiply it by  then adding the resulting value to in order to get the result ., Calculating relative error in Runga - Kutta's method from second and fourth order by knowing the analytical solution . The high efficiency of using Runga - Kutta's methods in solving the initial value problem comes through the numerical results obtained from the application of the methods and various examples.The new mathematical technique which we used in this study its an easy and accurate method that reduces errors and provide graphic solutions , so we recommended researcher to use it.