Solving Differential Equations For Equilibrium: Is a Different Answer Format Required? I'm going to include the exact wording of the question here, because I think it's relevant:
"First solve the equation $f(x)=0$ to find the critical points of the given autonomous differential equation $\frac{dx}{dt} = f(x)$. Then analyze the sign of $f(x)$ to determine whether each critical point is stable or unstable, and construct the corresponding phase diagram for the differential equation. Next, solve the differential equation explicitly for $x(t)$ in terms of $t$. Finally, use either the exact solution or a computer-generated slope field to sketch typical solution curves for the given differential equation, and verify visually the stability of each critical point."
$\frac{dx}{dt} = 3-x$
My Work
Now, this is a fairly straightforward problem, and I've had no issues apart from the format of my explicit solution. When I solve it, I perform:
$\frac{dx}{dt} = 3-x$
$\frac{1}{3-x}dx = dt$
$-ln(3-x) = t+C$
$x = 3-e^{C-t}$
$x = 3-e^Ce^{-t}$
$x = 3-Ce^{-t}$
Pretty routine stuff. However, if I look at the book's answer to this problem, I see:
$x = 3+(x_0 - 3)e^{-t}$
And I don't understand what I'm looking at. I haven't encountered any examples in the prep work that include $x_0$, and don't understand what $x_0$ represents. The given solution implies that $C = 3-x_0$, and I don't see how this happens. Would anyone be able to shed a bit of light on this?

Both are fine since $C$ is just arbitrary. Try experimenting with different values of $x_0$ and you'll see. When given the initial condition $x_0$ you will still be able to calculate $C$, so it wouldn't matter.
