Advanced Engineering Mathematics, 7th Edition

2.2 Separable Equations

INTRODUCTION

Consider the first-order equations dy/dx = f(x, y). When f does not depend on the variable y, that is, f(x, y) = g(x), the differential equation

(1)

can be solved by integration. If g(x) is a continuous function, then integrating both sides of (1) gives the solution y = ∫ g(x) dx = G(x) + c, where G(x) is an antiderivative (indefinite integral) of g(x). For example, if dy/dx = 1 + e^2x, then y = ∫ (1 + e^2x) dx or defined on (−, ).

A Definition

Equation (1), as well as its method of solution, is just a special ...