I'm confused with this procedure above. Why is this justified? Because $x(t) = e^{3t+c_1} + 1$ is the same thing as $x(t) = e^{3t}e^{c_1} +1$. Are they assuming that $e^{c_1}$ is just some constant scaling $e^{3t}$? If so, then shouldn't it be some new constant $c_2$? What sense is it making it the same as $c_1$? Because would that not imply that $e^{c_1} = c_1$ according to this answer?