Approximation of coupled differential equations with recursive integration

Question

I am currently reading the book "the quantum theory of light"
link: http://rplab.ru/~as/2000%20-%20R.Loudon%20-%20The%20Quantum%20Theory%20of%20Light%20-%203rd%20ed%20Oxford%20Science%20Publications.pdf
In the beginning of chapter 2.3 (page 52) is talked about an approximation method for the differential equations $$ Vcos(wt)e^{-iw_0t}C_2(t)=i\dot{C_1}(t)\\ Vcos(wt)e^{iw_0t}C_1(t)=i\dot{C_2}(t) $$ The approximation goes like this:

1. Insert the starting conditions into the left side of the equations. (here $C_1(0)=1$, $C_2(0)=0$)
2. Integrate the modified equations over time on both sides (from $0$ to $t$. So $\int_0^tdt$ on both sides). The result will be an expression for $C_1(t)$ and $C_2(t)$
3. Take the expressions and insert them into the left sides of the initial differential equations.
4. Integrate over time.....

If this process is done and infinitely then the result will be the exact solution of the differential equations.
My question now: Can this be proven?/ Has this been proven somewhere? I was trying to prove it myself but I was not quite able to. Is this maybe a common approximations for differential equations that is often used?