The aim of this paper is to introduce a symbolic technique for the computation of the solution to a complete ordinary differential equation with constant coefficients. The symbolic solution is computed via the variation of parameters method and, thus, constructed over the exponential matrix of the linear system associated with the homogeneous equation. This matrix is also symbolically determined. The accuracy of the symbolic solution is tested by comparing it with the exact solution of a test problem.
Perturbation theories for differential equations containing a small parameter
As explained in Henrard [
In order to achieve better accuracies in the applications of analytical theories, high orders of the approximate solution must be computed, making necessary a continuous maintenance and revision of the existing symbolic manipulation systems, as well as the development of new packages adapted to the peculiarities of the problem to be treated. Recently, Navarro [
One year later, Navarro [
As a first step, Navarro and Pérez [
The second step in the construction of the solution to (
This new symbolic package can be useful for research and educational purposes in the field of differential equations or dynamical systems. Nowadays there are many open problems which requires massive symbolic computation as, to cite one example, the analytical theory of the resonant motion of Mercury. We would like to stress that the aim of this work is to develop a symbolic tool, not a numeric method, as numeric solutions cannot be used in perturbation methods for differential equations. However, we have performed comparisons between the symbolic solution with a numeric one, just to show the efficiency of the technique.
In next section, we summarize the scheme proposed in [
As mentioned, Navarro and Pérez [
With the aid of the substitutions
Navarro and Pérez [
Thus,
As it has been stated in Section
if
In the following, we summarize some expressions which simplify the way in which the matrix
For any
For any
For any
For any
For the sake of simplicity, we have omitted the dependence on
The general solution to a nonhomogeneous linear differential equation of order
The computation of the solution to the constant coefficients linear part requires the calculation of the exponential of the matrix
In the following, we give a formula for the symbolic expansion of the solution to the complete ordinary differential equation. To that end, let us express the noncommutative parenthesis
Let us also express
In order to develop a symbolic expression of the solution to the complete differential equation, let us express
The product
Thus, (
The aim of this section is to illustrate the form that the approximation of the exponential matrix adopts. For that purpose, let us consider the differential equation given by
Taking, for instance,
Matrices
In Figure
Comparison of the solution computed through the symbolic method presented in this paper, and the numeric solution to the problem calculated by a RungeKutta fourth order method with a step of
In order to describe the algebraic processor, let us introduce the following test problem:
The first window shown by the symbolic processor allows us to introduce the parameters which define the differential equation to be solved, including the initial conditions.
The next window visualizes the expression of the companion matrix related to the ODE (Figure
Matrix
Then, matrices
Matrix
Matrix
Matrix
Matrix
In Figures
Parameters related to the accuracy of the solution to the differential equation.
Matrix
Comparison of real solution (black) and symbolic solutions for
Comparison of real solution (black) and symbolic solutions for
Table
Numerical evolution of the symbolic approximation
Step 



1  0.1  0.0951665336810 
2  0.2  0.1813305224920 
3  0.3  0.2594801299955 
4  0.4  0.3305856479131 
5  0.5  0.3955920028686 
6  0.6  0.4554122210115 
7  0.7  0.5109218189711 
8  0.8  0.5629540836788 
9  0.9  0.6122961986812 
10  1.0  0.6596861707192 
In Figure
Numerical error in the symbolic (




0.1  0.0000047853239  0.0000001672031 
0.4  0.0000712453497  0.0000358751267 
0.7  0.0002074165163  0.0003544605868 
1.0  0.0003710415717  0.0014307873617 
1.3  0.0004547161744  0.0038253117836 
1.6  0.0003284030320  0.0080436899562 
1.9  0.0000622406880  0.0144197157174 
2.2  0.0006216534547  0.0230329551851 
2.5  0.0011277539714  0.0336675456541 
2.8  0.0013572764337  0.0458134712942 
3.1  0.0012347982225  0.0587076431208 
3.4  0.0008953272672  0.0714078077594 
3.7  0.0006002231245  0.0828896894497 
4.0  0.0005492334199  0.0921556872924 
4.3  0.0007154938930  0.0983431460014 
4.6  0.0008300921248  0.1008204330678 
4.9  0.0005487250652  0.0992608551151 
5.2  0.0002963137375  0.0936868279861 
5.5  0.0015298326576  0.0844798593631 
5.8  0.0026838757810  0.0723553936143 
6.1  0.0032571351779  0.0583046260627 
6.4  0.0030289419613  0.0435091879952 
6.7  0.0022189180928  0.0292364988851 
7.0  0.0013691340100  0.0167257096706 
7.3  0.0010043530807  0.0070750555622 
7.6  0.0012821439144  0.0011409176473 
7.9  0.0018689726696  0.0005417339524 
8.2  0.0021436005756  0.0021897330323 
8.5  0.0016148602301  0.0091082291225 
8.8  0.0002869277587  0.0196147078451 
9.1  0.0012796703448  0.0327897203523 
9.4  0.0022771057968  0.0474740017777 
9.7  0.0021518593816  0.0623710756305 
10.0  0.0009803998394  0.0761625070468 
Exact solution (black), symbolic solution (blue), and numeric solution (green).
Error in the symbolic (black) and numeric (blue) approximations.
We have developed a symbolic processor as well as a symbolic technique in order to deal with the solution to a linear ordinary differential equation with constant coefficients of order