How to rewrite exponential function $e^{x+c}$ into $ce^x$?

Question

Tried to rewrite $e^{x+c}$ by using exponential properties $e^x.e^c=e^{x+c}$ but struggle to understand how to get into $ce^x$. Is it always the case that $c=e^c$?

Edit: To put in context, I solved a separable differential equation $\frac{dy}{dx}=y$, I get the solution is $y = e^{x+c}$. But the solution in the book is written $y=ce^x$.

Answer 1

It truly is :

$$e^{x+c} = e^c \cdot e^x$$

Now, if you simply manipulate the constant $c$ to be $c := e^c$ the expression becomes the desired :

$$\boxed{e^{x+c} = ce^x}$$

Answer 2

Hint

If $c$ is a constant then $e^c$ is still a constant.

Note that in mathematics constant are arbitrary: if a call $c$ a constant and then take $c+1$ I can still call it $c$ (note that this doesn't mean that they are the same constant, but nontheless both of them are a constant)