Linear Differential Equation - Introduction

Introduction

Linear differential equations are of the form

where the differential operator L is a linear operator, y is the unknown function (such as a function of time y(t)), and the right hand side ƒ is a given function of the same nature as y (called the source term). For a function dependent on time we may write the equation more expressly as

and, even more precisely by bracketing

The linear operator L may be considered to be of the form

L_n(y) \equiv \frac{d^n y}{dt^n} + A_1(t)\frac{d^{n-1}y}{dt^{n-1}} + \cdots + A_{n-1}(t)\frac{dy}{dt} + A_n(t)y \,

The linearity condition on L rules out operations such as taking the square of the derivative of y; but permits, for example, taking the second derivative of y. It is convenient to rewrite this equation in an operator form

where D is the differential operator d/dt (i.e. Dy = y', D2y = y",... ), and the An are given functions.

Such an equation is said to have order n, the index of the highest derivative of y that is involved.

A typical simple example is the linear differential equation used to model radioactive decay. Let N(t) denote the number of radioactive atoms in some sample of material at time t. Then for some constant k > 0, the number of radioactive atoms which decay can be modelled by

If y is assumed to be a function of only one variable, one speaks about an ordinary differential equation, else the derivatives and their coefficients must be understood as (contracted) vectors, matrices or tensors of higher rank, and we have a (linear) partial differential equation.

The case where ƒ = 0 is called a homogeneous equation and its solutions are called complementary functions. It is particularly important to the solution of the general case, since any complementary function can be added to a solution of the inhomogeneous equation to give another solution (by a method traditionally called particular integral and complementary function). When the Ai are numbers, the equation is said to have constant coefficients.

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