Positioning of Constant C in Antiderivative

Question

Standard notation would just include a $C$ as constant in the end, as below:

$y = f(x) + C$

however I see quite a few tutorials/text books that just use

$y = Cf(x)$, especially when $ln()$ is encountered.

How can we transform the standard notation to the notation where $C$ is a coefficient of the $f(x)$?

Answer 1

I think you are generally mistaken about alternate forms. One does indeed usually have an additive constant. However, you can sometimes manipulate the result into something different.

For example, when you solve $y' = y$, you get a separable differential equation $$\frac{dy}{dx} = y$$ which separates into $$\frac{dy}{y} = dx.$$ Now, integrating both sides you get $$ \ln y + C_1 = x + C_2, $$ where $C_1$ and $C_2$ are arbitrary constants. Then, so is $K = C_2-C_1$ and we can write $$ \ln y = x + K $$ and exponentiate both sides to get $$ y = e^{x+K} = e^K e^x = Ce^x, $$ where $$C = e^K = e^{C_2-C_1}.$$

As you can see, $C$ is just a algebraic modification of the original additive constants.

Answer 2

If we have $\ln y=f(x)+C_0$, then we exponentiate to find an expression for $y$ itself:

$$y=e^{f(x)+C_0}=e^{C_0}e^{f(x)}=CF(x),$$

where we are taking $C=e^{C_0}$ and $F(x)=e^{f(x)}$.

In short, exponentiation turns addition into multiplication, so an added constant becomes a multiplied coefficient.

Answer 3

Writing $y=f(x)+C$ is a somewhat lousy way of denoting the set of functions $x\mapsto f(x)+C$, where $f$ is a given function and $C$ is an arbitrary real constant. Similarly, writing $y=C f(x)$ is a somewhat lousy way of denoting the set of functions $x\mapsto C\,f(x)$, where $f$ is a given function and $C$ is an arbitrary real constant. Depending on the context solution sets of certain problems, e.g., finding the set of all primitives of a function $g$, are of one of these kinds.