2.1 Introduction

Consider the two-dimensional linear system

where a, b, c, d are real numbers. Depending on the nature of eigenvalues of the coefficient matrix , we can construct a non-singular real matrix C such that the reduces the system (2.1.1) into one of the following systems:

• (i) u′ = λu, v′ = μv

• (ii) u′ = λu + v, v′ = λv

• (iii) u′ = αu + βv, v′ = −βu + αv.

Here λ and μ are the eigenvalues of A, in case A has real eigenvalues; they may be equal. Whereas α and β(≠ 0) are respectively the real and the imaginary parts of the eigenvalues, in case A has non-real eigenvalues (they occur in conjugate pair) α ± iβ. It is easy to write down the solutions of the reduced systems. We have u(t) = u0eλt, v(t) = v0eμt for the case (i). The solutions are u(t) = (u0 + v0t)eλt, v(t) = v0eλt for the case (ii). Finally, for the case (iii), we have u(t) = eαt (u0 cos(βt) + v0 sin(βt)), v(t) = eαt(v0 cos(βt) − u0 sin(βt)). Using the matrix C, we can then write down the expressions for the solutions x, y.

The same procedure works for higher-dimensional linear systems. However, it is not easy to construct the matrix C as it involves deeper aspects of the eigenvalues and the corresponding eigenvectors of the coefficient matrix A and involves the Jordan canonical form. We only remark that each component of the solution of the linear system x′ = Ax is a linear combination of the products of the functions of t, and these functions are polynomials, exponentials and (in the presence of complex eigenvalues) the trigonometric functions–sine and cosine functions.