##### TI-Nspire CAS in Engineering Mathematics: First Order Systems and Symbolic Matrix Exponentiation

Solving Ordinary Differential Equations by using a library of Laplace Transformations

Publisher: T³ Europe

Author: T³ Europe, Michel Beaudin

Topic: MINT

Tags Ingenieurswissenschaften , Matrix

The Library of Laplace Transforms ("ETS_specfunc.tns") contains the function called "simultd" that can be used to solve a system of ODEs, using Laplace transforms techniques. Generally speaking, a first order, constant coefficient system, has the following form:

where A is a square (n × n) matrix with real entries, g(t) is a n × 1 matrix (column vector), t₀ ∈ ℝ and x₀ is a n × 1 matrix (constant column vector). Solving the system means to find the n × 1 matrix (column vector) X(t) that satisfies both the differential system and the initial condition.

A is a matrix and not scalar... But TI-Nspire CAS knows how to compute "the exponential of a matrix". Our goal is to define, IN EXACT MODE (when possible), the exponential of a square matrix A: e(At) (t a real variable).

We make use of the library ETS_specfunc.tns to easily compute e(At); applying our findings in a concrete example of a system originating from connected salt tanks.

And finally we apply our solution of to the problem of finding parametric equations to a great circle.

To be able to use the tns file below, you need to save "ETS_specfunc.tns" in the TI-Nspire CAS "Mylib" folders and refresh your libraries.