1.1 Introduction

The first-order equations–linear and non-linear–and the second-order linear equations, with constant or variable coefficients, are considered in this chapter. For solving the first-order equations, familiar methods such as the method of separation of variables or a method that can be reduced to this are used. Also discussed are the exact differential equations or those equations that can be reduced to this form using a suitable integrating factor (IF). We also emphasize the peculiarities that may arise in an initial value problem (IVP) when sufficient conditions imposed in the Cauchy–Peano existence theorem or in the method of Picard's iterations are not satisfied. Many exercises deal with the maximal interval of existence of a solution to an IVP.

Only second-order linear equations are considered here. The non-linear equations or, more generally, the two-dimensional systems of first-order equations are treated in Chapter 5 on qualitative analysis. The treatment of equations with constant coefficients is straightforward. The equations with variable coefficients are more difficult to deal with, and, in general, it is not possible to obtain the solution in explicit form. However, the structure of solutions to the homogeneous and inhomogeneous equations is well-understood.

A general first-order ordinary differential equation (ODE) takes the form f(t, x(t), x′(t)) = 0, where f is a given function and x = x(t) is the unknown function to be determined. A general theory for the above equation is rather difficult.