### 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) ...

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