Analytical and Numerical Solution Analysis of Legendre Differential Equation

  • Hadi Susanto Universitas Negeri Semarang, Indonesia
  • St. Budi Waluya Universitas Negeri Semarang, Indonesia


In general, differential equations can be solved analytically and numerically. One of the important equations is Legendre differential equation, the general form  where n is a constant. This research aimed to determine the form of analytical and numerical solutions, and numerical simulation solutions of Legendre differential equation. Analytic solution by power series method produces y(x) in the form of power series, where To minimize errors because of cutting series in special solutions of Legendre differential equation, the results obtained are converted into the series containing   where the series is equivalent to  The value of and  are obtained by substituting the initial value which is known in the general solution of the differential equations. This research used value of  In general, it can be concluded that in a special solution of Legendre differential equation, and are applied. For the same step size, the error solution of Legendre differential equation by fourth order Adams-Bashforth-Moulton method is more than error fourth order Runge-Kutta method. In general, it can be concluded that the fourth order Runge-Kutta method in the numerical solution of Legendre differential equation is more accurate than fourth order Adams-Bashforth-Moulton method.