Page:Calculus Made Easy.pdf/263

Is this an exact differential or not? Apply the test.

$\left\{{\begin{aligned}{\frac {d(1+3xy)}{dy}}=3x,\\{\dfrac {d(x^{2})}{dx}}=2x,\end{aligned}}\right.$

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

$(2x+6x^{2}y)\,dx+2x^{3}\,dy=0$.

Now apply the test to this.

$\left\{{\begin{aligned}{\frac {d(2x+6x^{2}y)}{dy}}=6x^{2},\\{\dfrac {d(2x^{3})}{dx}}=6x^{2},\end{aligned}}\right.$

which agrees. Hence this is an exact differential, and may be integrated. Now, if $w=2x^{3}y$,

$dw=6x^{2}y\,dx+2x^{3}\,dy$.

Hence

$\int 6x^{2}y\,dx+\int 2x^{3}\,dy=w=2x^{3}y$;

so that we get

$U=x^{2}+2x^{3}y+C$.