Algebraic Solution to Simultaneous Linear First-Order Non-Homogeneous Differential Equations with Constant Coefficients

Abstract

Ordinary differential equations, both first and second order, are essential in the modeling of many physical systems. A system of simultaneous differential equations results from more complicated modeling involving more than one dependent variables with respect to a single independent variable. There are several methods in solving a system of simultaneous linear differential equations including variable substitution, Laplace transform and using the D-operator.
Proposed in this paper is a simplified method of solving a set of two non-homogeneous linear first-order simultaneous ordinary differential equations with constant coefficients that falls into a certain form. An algebraic formula is developed to compute the solution to the said differential equations provided a certain necessary condition is satisfied.
Four different forms of the functions of the independent variable on the right side of the equations, namely constants, linear functions, natural exponential functions and sinusoidal functions, are considered. For each case, an algebraic formula to calculate the dependent variable as well as its derivative are developed. Moreover, these simple algebraic formulae can be easily programmed into spreadsheet where one just has to enter the values of the constants and coefficients from the original equations and instantly obtain the correct answers.