Solving Differential Equations

Just a quick question; when solving differential equations using separation of variables, when we integrate both sides, how come it's only plus C to one of the sides and not both sides?

Thanks!

Let's say your answer ends up
(1) y + C_1 = x + C_2,

which is the same as
(2) y = x + C_2 - C_1.

But recall C_1 and C_2 are just arbitrary constants (the key being arbitrary). They can just merge into one C.
That said, is it wrong to leave your answer in (1)? Technically no, but you can't do initial value problem with that.

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It's kind of like if you end up with
ln(y) = x + C,
and you rewrite it as
y=Ae^x. The "A" is "e^C", which itself is arbitrary.