First order linear differential equations

Abstract

A differential equation is an equation involving an unknown function and its derivatives. By a solution of a differential equation we mean a function that is differentiable and satisfies the equation on some interval. For example, x′ − x = 0 is a differential equation involving an unknown function x and its first derivative with respect to an independent variable that we may call t, s, etc. We notice that (e ^t)′ −e ^t = e ^t−e ^t = 0 for all t in the interval I = (−∞; ∞). Therefore, x(t) = e ^t is a solution of the differential equation on the interval I.