ABSTRACT
Starting with a pair of coupled first order equations, solution set (along with phase portraits) is obtained using eigenvalues and eigenvectors with attention given to defective coefficient matrices. Additionally, the pair is converted to a second order homogeneous differential equation with the solutions obtained using the characteristic equation and further verified using Laplace transforms. Examples are annotated with theory, comparison of results and verification using Runge-Kutta methods. The analysis is extended to a pair of non-homogeneous first order equations with particular solutions obtained using the method of variation of parameters. The solution set is validated through the use of Laplace transforms and Runge-Kutta methods. The multiple coupled first order non-homogeneous systems are analyzed next. The eigenvalue based approach is limited to four equations because of the complexity associated with defective matrices. Laplace transform based solution has been used as a means for verification and for sets with 5 or more equations, only the Laplace transform method is used. In every case, the solution is verified using the Runge-Kutta method. The examples shown cover a wide array of possibilities with each example fully annotated with the appropriate theory, explanations, justifications and even verification carried out symbolically and displayed.
