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