Page:Calculus Made Easy.pdf/263

 Is this an exact differential or not? Apply the test. {{c|$$\left\{\begin{aligned} \frac{d(1 + 3xy)}{dy}=3x, \\ \dfrac{d(x^2)}{dx} = 2x, \end{aligned}\right.$$}} which do not agree. Therefore, it is not an exact differential, and the two functions $$1+3xy$$ and $$x^2$$ have not come from a common original function.

It is possible in such cases to discover, however, an integrating factor, that is to say, a factor such that if both are multiplied by this factor, the expression will become an exact differential. There is no one rule for discovering such an integrating factor; but experience will usually suggest one. In the present instance $$2x$$ will act as such. Multiplying by $$2x$$, we get

Now apply the test to this. {{c|$$\left\{ \begin{aligned} \frac{d(2x + 6x^2y)}{dy}=6x^2, \\ \dfrac{d(2x^3)}{dx} = 6x^2, \end{aligned} \right.$$}} which agrees. Hence this is an exact differential, and may be integrated. Now, if $$w = 2x^3y$$,

Hence

so that we get