Symbolically Precise Solutions to a Homogeneous Second Order Matrix Ordinary Differential Equation with Macsyma
✍ Scribed by Harry S.D. Mills
- Publisher
- Elsevier Science
- Year
- 1993
- Tongue
- English
- Weight
- 230 KB
- Volume
- 15
- Category
- Article
- ISSN
- 0747-7171
No coin nor oath required. For personal study only.
✦ Synopsis
A batch file tube_ode2, written in Macsyma version 309.6 for the SUN 3/60 is presented, which uses Laplace transform theory to solve the homogeneous second order matrix ordinary differential equation (F^{\prime \prime}(t)+R F(t)=0), where (F(t)) is an (n) by (n) matrix with entries that are infinitely differentiable functions of the real parameter (t, F^{\prime \prime}(t)) is the second derivative of (F(t)) with respect to (t), and (R) is an (n) by (n) matrix with constant entries. The batch file accepts (R), and the initial conditions (F(0)) and (F^{\prime}(0)) as matrices, and the output is a matrix with functional entries.
It is shown that if (R, F(0)), and (F^{\prime}(0)) have rational entries, symbolically precise solutions are obtainable in all cases up to (n=4). Practice indicates that symbolically precise solutions are often obtainable for (n>4), as well. To accomplish this, the method of the adjoint is used in solving the matrix equation (A L(F(t))=P) arising from the action of the Laplace transform on the given ODE, from which the final solution (F(t)) is computed as the inverse transform of (\boldsymbol{L}(F(t))). Macsyma's ilt function is utilized in this regard, with a boost from the user-defined function force_factor, which expresses the determinant of (A) as the product of quadratic factors.