Condition for Exactness

Let $D$ be a region in $xy$ -plane and let $M$ and $N$ be real valued functions defined on $D.$ Consider an equation

$$M(x, y(x)) dx + N(x, y(x)) dy = 0, \; (x, y(x)) \in D.$$ (15.6.9)

DEFINITION 15.6.1 (Exact Equation)   The Equation (14.6.9) is called Exact if there exists a real valued twice continuously differentiable function $f$ such that

$$\frac{\partial f}{\partial x} = M \; {\mbox{ and }} \; \frac{\partial f}{\partial y} = N.$$

THEOREM 15.6.2   Let $M$ and $N$ be ``smooth" in a region $D.$ The equation (14.6.9) is exact if and only if

$$\frac{\partial M}{\partial y}= \frac{\partial N}{\partial x}.$$ (15.6.10)

Proof. Let Equation (14.6.9) be exact. Then there is a ``smooth" function $f$ (defined on $D$ ) such that $M = \frac{\partial f}{\partial x}$ and $N = \frac{\partial f}{\partial y}.$ So, $\frac{\partial M}{\partial y}= \frac{\partial^2 f}{\partial y \partial x} = \frac{\partial^2 f}{\partial x \partial y}= \frac{\partial N}{\partial x}$ and so Equation (14.6.10) holds.

Conversely, let Equation (14.6.10) hold. We now show that Equation (14.6.10) is exact. Define $G(x,y)$ on $D$ by

$$G(x,y) = \int M(x,y) dx + g(y)$$

where $g$ is any arbitrary smooth function. Then $\frac{\partial G}{\partial x} = M(x,y)$ which shows that

$$\frac{\partial}{\partial x} \cdot \frac{\partial G}{\partial y} =... ...}{\partial x} = \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}.$$

So $\frac{\partial }{\partial x} (N - \frac{\partial G}{\partial y}) = 0$ or $N - \frac{\partial G}{\partial y}$ is independent of $x.$ Let $\phi(y) = N - \frac{\partial G}{\partial y}$ or $N = \phi(y) + \frac{\partial G}{\partial y}.$ Now

$$M(x,y) + N \frac{dy}{dx} = \frac{\partial G}{\partial x} + \left[ \frac{\partial G}{\partial y} + \phi(y)\right] \frac{dy}{dx}$$

$$= \left[ \frac{\partial G}{\partial x} + \frac{\partial G}{\partial... ...rac{dy}{dx} \right] + \frac{d}{dy} \left( \int \phi(y) dy \right) \frac{dy}{dx}$$

$$= \frac{d}{dx} G(x, y(x)) + \frac{d}{dx}\left( \int \phi(y) dy \right) \hspace{.5in} {\mbox{where }} y = y(x)$$

$$= \frac{d}{dx} \bigl( f(x,y) \bigr) \hspace{.5in} {\mbox{ where }} f(x,y) = G(x,y) + \int \phi(y) dy$$

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