Comparative Analysis of Approximate Solutions of Differential Equations Using Euler and Runge-kutta Methods
Sima Pashayeva 
Abstract. The article presents a comparative analysis of non-quadratic solutions of differential equations, that is, numerical solutions and quadrature solutions. In the problem under consideration, the results obtained from an approximate solution of a given differential equation using the Euler and Runge-Kutta methods were compared with the results obtained from an exact solution of this equation. The solution to the problem is presented in analytical, tabular and graphical form. The analysis of the results shows that the accuracy of the solution of the problem using the Euler method is relatively higher than the accuracy of the solution using the Runge-Kutta numerical method. This explains the correct division of a given segment in the numerical solution of the problem and the accuracy of the choice of steps in this division. At each precisely chosen step, the intersection points of the lines and the angular coefficients of the tangents drawn to the integral curve at these points were found, which shows that the direction of the tangent at any point of the corresponding increment of the function and the integral curve coincides with the direction field.
Keywords: Leonard Euler method, Runge-Kutta method, integral curve, increment ratio, integral curve, broken line scheme