I have never encountered a negative sign in the solution while solving differential equations. Therefore, I was tricked that I have done something wrong. (No one told me before.)
So blame whoever taught you! (I didn't blame you, did I? But I was
surprised, for good reason, that you would not be familiar with this.)
Here is one example of what I referred to, so you can see at least that it is taught somewhere:
We now examine a solution technique for finding exact solutions to a class of differential equations known as separable differential equations. These equations are common in a wide variety of …
math.libretexts.org
In the first example, it says,
Now we use some logic in dealing with the constant [imath]C[/imath]. Since [imath]C[/imath] represents an arbitrary constant, [imath]3C[/imath] also represents an arbitrary constant. If we call the second arbitrary constant [imath]C_1[/imath], where [imath]C_1=3C[/imath],
the equation becomes
[math]\ln|3y+2|=x^3−12x+C_1.[/math]
Now exponentiate both sides of the equation (i.e., make each side of the equation the exponent for the base e).
[math]e^{\ln|3y+2|}=e^{x^3−12x+C_1}\\|3y+2|=e^{C_1}e^{x3−12x}[/math]
Again define a new constant [imath]C_2=e^{C_1}[/imath] (note that \(C_2>0):
[math]|3y+2|=C_2e^{x3−12x}.[/math]
And so on. We constantly rename constants.
Here's another source that says more, first about
negatives in particular:
Before equation (7), he says:
Now, from a notational standpoint we know that the constant of integration, c, is an unknown constant and so to make our life easier we will absorb the minus sign in front of it into the constant and use a plus instead. This will NOT affect the final answer for the solution.
Again, after equation (7), it says,
There is a lot of playing fast and loose with constants of integration in this section, so you will need to get used to it. When we do this we will always to try to make it very clear what is going on and try to justify why we did what we did.
In another place, he talks about similar manipulation, in the context of
integrals:
In the first example, he says,
Now, both c and k are unknown constants and so the sum of two unknown constants is just an unknown constant and we acknowledge that by simply writing the sum as a c.
In the second example:
However, since the constant of integration is an unknown constant dividing it by 2 isn’t going to change that fact so we tend to just write the fraction as a c.