Symbolically Precise Solutions to a Homogeneous Second Order Matrix Ordinary Differential Equation with Macsyma

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″(t) + RF (t) = 0, where F(t) is an n by n matrix with entries that are infinitely differentiable functions of the real parameter t, F″(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′(0) as matrices and the output is a matrix with functional entries. It is shown that if R, F(0), and F′(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 (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 (F(t)). Macsymaʹsilt 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.