ABSTRACT
Chapters 1 to 4 include the development of vectors, determinants and linear algebra and their properties. Chapter 7 considers the elements of calculus of matrices. This base leads us to think of the development of a system of differential equations. They are of two types (i) linear systems and (ii) non-linear systems of DES. This chapter includes the main feature of the first type.
Suppose that a particle is floating in space freely, it's position at time, t, in space is described by its co-ordinates in space i.e (x(t), y(t), z(t)). Where ‘t’ is a scalar. The path of the particle involves the time derivative of the vector, namely, (x'(t), y'(t), z'(t)). Here the motion of the particle is a coupled motion. The wind factor also may change its path. There are several problems in real -life-situations which involve coupled action between its variables and demand their solutions.
Once linear systems are modelled, one needs to know if a solution exists and is unique. In case there is a perturbation (a boosting effect), can one think of controlling the behaviour of the elements of the system as ‘t’ increases? How to find a solution of such a system? At this stage, knowledge of vectors, linear algebra and elements of calculus play vital role. The plan of the Chapter 8 is as follows.
In Section 8.2 linear systems of differential equations are introduced and their solutions called 'fundamental matrices' (FM) form the content of Section 8.3. Method of successive approximations, an approach to obtain the solution is given in Section 8.4. Section 8.5 considers nonhomogeneous systems and Section 8.6 deals with linear systems of differential equations with constant coefficients
Perturbations in a system are then added to determine the nature of solutions. The behaviour of solutions in time - dimension known as `stability' is also considered and this is the content of Section 8.7. Lastly, in Section 8.8 an application, `Election Mathematics' which forms a part of game theory, has been included.
