Page:Calculus Made Easy.pdf/256

where $\log _{\epsilon }C$ is the yet undetermined constant ^[1] of integration. Then, delogarizing, we get:

$y=C\epsilon ^{-{\frac {a}{b}}x}$,

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 ^[2] of $y$ at starting. This we may call $y_{0}$, and so write the solution as

$y=y_{0}\epsilon ^{-{\frac {a}{b}}x}$.

 

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Example 2.
Let us take as an example to solve

$ay+b{\frac {dy}{dx}}=g$,

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

1. We may write down any form of constant as the “constant of integration,” and the form $\log _{\epsilon }C$ is adopted here by preference, because the other terms in this line of equation are, or are treated as logarithms; and it saves complications afterward if the added constant be of the same kind.
2. Compare what was said about the “constant of integration,” with reference to Fig. 48 on p. 187, and Fig. 51 on p. 190.