# Time-dependent perturbation theory derivation with 2-level system (Griffiths)

I'm reviewing time-dependent perturbation theory (TDPT) via Griffiths QM book. I'm looking at section 9.12: he reviews the results for a two-level system where the system starts in one eigenstate, ie. $c_a(t=0) = 1$, $c_b(t=0)=0$ [9.15]. To zeroth order, then, $c_a^{(0)}(t) = 1$, $c_b^{(0)}(t)=0$. Plugging in the value for $c_b$ into the RHS of $\frac{d}{dt}c_a^{(1)} = -i/\hbar H'_{ab} e^{-i\omega_0 t} c_b^{(0)}$ (where $H'_{ab}$ is the matrix element connecting states a and b) one gets $\frac{d}{dt} c_a^{(1)} = 0$ [9.17], which Griffiths then says leads to $c_a^{(1)}(t)=1$. My question is about the very last part. Why do we get $c_a^{(1)}(t)=1$ instead of some other constant $C<=1$? I can't see a constraint in the problem which makes the result 1 rather than some arbitrary constant.

Notice that in my notation, $c_a^{(2)}(t)$ includes the zeroth order term $[\ldots]$
Because $c_a(0)=1$ and $c_b(0)=0$, the correction term of every order is zero when $t=0$. They will generically become non-zero as $t$ grows, but to first order, $\dot c_a^{(1)}=0$, which means that $c_a^{(1)}(t) = c_a^{(0)}(t) = 1$.