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**Unformatted text preview: **€² âˆ« y dx = âˆ« 1 dx .
Although we don't know the function y that appears in the integrand on the left side, we can use
the general integration formula discussed in Chapter 1 for functions u of x , âˆ« uâ€² / u dx = ln u + C , to evaluate the left side. The right side can be explicitly integrated. We get
ln y + C = x + D .
Since C and D are arbitrary constants, we combine them on the right side as D âˆ’ C obtaining the
so-called implicit solution of the differential equation,
ln y = x + D âˆ’ C . (2.2) The constant D âˆ’ C on the right is again an arbitrary constant. In order to avoid a proliferation of
letters, we will simply call this C again, as we have no interest in keeping track of the original
value of all the constants. In general, in these types of manipulations, if an arbitrary constant is
modified to produce a constant of a slightly different form, we will usually continue labeling the
new variant with the same letter. Note that here we could have avoided this notational annoyance
had we only used a single constant of integration in deriving (2.2). In the future we shall include
only one constant, on the right side of the integrated equation.
We must now solve the implicit equation (2.2) for y . In order to extract the y term on the left we
use the identity eln w = w , valid for any positive w . This identity is simply a restatement of the
inverse function relationship that holds between the exponential and logarithm functions. Thus
from (2.2) (with D âˆ’ C replaced by C ) we get
y =e ln y = e x + C = eC e x . Since y = Â± y , we can write the last equation as y = Â± eC e x . The quantity Â± eC is just a constant,
so following the convention mentioned earlier, we again denote its value by C . Thus we obtain
the final solution y = Ce x . !
We can simplify the mechanics of the solution process by using the dy / dx form of the differential
equation. In this procedure we not only separate the x and y terms, but we also separate the dx
and dy differentials, placing each with the correspondin...

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