Advanced Engineering Mathematics, 7th Edition

2.4 Exact Equations

INTRODUCTION

Although the simple differential equation y dx + x dy = 0 is separable, we can solve it in an alternative manner by recognizing that the left-hand side is equivalent to the differential of the product of x and y; that is, y dx + x dy = d(xy). By integrating both sides of the equation we immediately obtain the implicit solution xy = c.

Differential of a Function of Two Variables

If z = f(x, y) is a function of two variables with continuous first partial derivatives in a region R of the xy-plane, then its differential (also called the total differential) is

(1)

Now if f(x, y) = c, it follows from (1) that

(2)