Essential Ordinary Differential Equations

A linear, nth order (where n is a positive integer), ordinary differential equation in standard form is the problem y(n) + pn−1(x)y(n−1) +···+ p1(x)y + p0(x)y = g(x), (1.1) where the coefficient functions1 p0(x), ..., pn−1(x) and the function g(x) are defined and real valued in an open interval I of the real line R. The interval I can be bounded or unbounded. The function g is called the inhomogeneous term. If g = 0 (that is g is the zero function taking the value 0 for all values of its argument x) then we say that the problem is homogeneous. Otherwise it is called non-homogeneous or inhomogeneous. A solution of (1.1) is an n-times differentiable function y(x) defined in the interval I , such that dny dxn (x) + pn−1(x) dn−1y dxn−1 (x) +···+ p1(x) dy dx (x) + p0(x)y(x) = g(x) (1.2) for all x in I . The different components of (1.1) have conventional names. The variable x is called the independent variable. The variable y is called the dependent variable; we 1 We do not aim for consistency in the way we talk about functions, sometimes following the custom of practical calculus to speak of “the function f (x)”. Otherwise we speak of “the function f ”, especially if ambiguity may arise, for example if we wish to assert that f is zero.