9781107697492book CUP/SCHL3 August 25, 2011 19:15 Page-1 1 First order di↵erential equations 1.1 General remarks about di↵erential equations 1.1.1 Terminology A di↵erential equation is an equation involving a function and its derivatives. An example which we will study in detail in this book is the pendulum equation d2x = sin(x), (1.1) dt2 − which is a di↵erential equation for a real-valued function x of one real variable t. The equation expresses the equality of two functions. To make this clear we could write (1.1) more explicitly as d2x (t)= sin(x(t)) for all t R, dt2 − 2 but this would lead to very messy expressions in more complicated equations. We therefore often suppress the independent variable. When formulating a mathematical problem involving an equa- tion, we need to specify where we are supposed to look for solu- tions. For example, when looking for a solution of the equation x2 + 1 = 0 we might require that the unknown x is a real number, in which case there is no solution, or we might allow x to be com- plex, in which case we have the two solutions x = i. In trying ± to solve the di↵erential equation (1.1) we are looking for a twice- di↵erentiable function x : R R. The set of all such functions is ! very big (bigger than the set of all real numbers, in a sense that can be made precise by using the notion of cardinality), and this 1 9781107697492book CUP/SCHL3 August 25, 2011 19:15 Page-2 2 First order di↵erential equations is one basic reason why, generally speaking, finding solutions of di↵erential equations is not easy. Di↵erential equations come in various forms, which can be clas- sified as follows. If only derivatives of the unknown function with respect to one variable appear, we call the equation an ordinary di↵erential equation, or ODE for short. If the function depends on several variables, and if partial derivatives with respect to at least two variables appear in the equation, we call it a partial di↵eren- tial equation, or PDE for short. In both cases, the order of the di↵erential equation is the order of the highest derivative occur- ring in it. The independent variable(s) may be real or complex, and the unknown function may take values in the real or complex numbers, or in Rn of Cn for some positive integer n. In the lat- ter case we can also think of the di↵erential equation as a set of di↵erential equations for each of the n components. Such a set is called a system of di↵erential equations of dimension n. Finally we note that there may be parameters in a di↵erential equation, which play a di↵erent role from the variables. The di↵erence be- tween parameters and variables is usually clear from the context, but you can tell that something is a parameter if no derivatives with respect to it appear in the equation. Nonetheless, solutions still depend on these parameters. Examples illustrating the terms we have just introduced are shown in Fig. 1.1. In this book we are concerned with ordinary di↵erential equa- tions for functions of one real variable and, possibly, one or more parameters. We mostly consider real-valued functions but complex- valued functions also play an important role. 1.1.2 Approaches to problems involving di↵erential equations Many mathematical models in the natural and social sciences in- volve di↵erential equations. Di↵erential equations also play an important role in many branches of pure mathematics. 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