Differential Equations and Linear Algebra, 4th Edition

1.2 Integrals as General and Particular Solutions

The first-order equation $dy/dx=f(x,y)$ takes an especially simple form if the right-hand-side function f does not actually involve the dependent variable y, so

$${\displaystyle \frac{dy}{dx}}=f(x).$$ (1)

In this special case we need only integrate both sides of Eq. (1) to obtain

$$y(x)={\displaystyle \int f(x)dx+C.}$$ (2)

This is a general solution of Eq. (1), meaning that it involves an arbitrary constant C, and for every choice of C it is a solution of the differential equation in (1). If G(x) is a particular antiderivative of f—that is, if $G$