Page:Calculus Made Easy.pdf/256

 where $$\log_\epsilon C$$ is the yet undetermined constant of integration. Then, delogarizing, we get:

which is the solution required. Now, this solution looks quite unlike the original differential equation from which it was constructed: yet to an expert mathematician they both convey the same information as to the way in which $$y$$ depends on $$x$$.

Now, as to the $$C$$, its meaning depends on the initial value of $$y$$. For if we put $$x=0$$ in order to see what value $$y$$ then has, we find that this makes $$y = C \epsilon^{-0}$$; and as $$\epsilon^{-0} = 1$$ we see that $$C$$ is nothing else than the particular value of $$y$$ at starting. This we may call $$y_0$$, and so write the solution as

Example 2. Let us take as an example to solve

where $$g$$ is a constant. Again, inspecting the equation will suggest, (1) that somehow or other $$\epsilon^x$$ will come into the solution, and (2) that if at any part of the