Page:Calculus Made Easy.pdf/266

 Subtracting (2) from (1) and dividing by $$2$$, we then have

which is more conveniently written

Or, the solution, which at first sight does not look as if it had anything to do with the original equation, shows that $$y$$ consists of two terms, one of which grows logarithmically as $$x$$ increases, and of a second term which dies away as $$x$$ increases.

Example 7.

Let

Examination of this expression will show that, if $$b=0$$, it has the form of Example 1, the solution of which was a negative exponential. On the other hand, if $$a=0$$, its form becomes the same as that of Example 6, the solution of which is the sum of a positive and a negative exponential. It is therefore not very surprising to find that the solution of the present example is

where

The steps by which this solution is reached are not given here; they may be found in advanced treatises.