Page:Calculus Made Easy.pdf/262

 Example 4.

Suppose that $$\quad \quad \quad M\, dx + N\, dy = 0$$.

We could integrate this expression directly, if $$M$$ were a function of $$x$$ only, and $$N$$ a function of $$y$$ only; but, if both $$M$$ and $$N$$ are functions that depend on both $$x$$ and $$y$$, how are we to integrate it? Is it itself an exact differential? That is: have $$M$$ and $$N$$ each been formed by partial differentiation from some common function $$U$$, or not? If they have, then {{c|$$\left\{ \begin{aligned} \frac{\partial U}{\partial x} = M, \\ \frac{\partial U}{\partial y} = N. \end{aligned} \right.$$}} And if such a common function exists, then

is an exact differential (compare p. 175).

Now the test of the matter is this. If the expression is an exact differential, it must be true that

for then

which is necessarily true.

Take as an illustration the equation