Set of ODE: how can I solve it?I want to solve this system, but I...

After putting solution of first equation in 2nd, we get,
$\frac{dB}{dt}+bB=a{A}_0{e}^{-at}$
This is a linear differential equation in $B$ which can be solved by multiplying integrating factor $(e^{bt})$ on both sides.
EDIT:
$e^{bt}\frac{dB}{dt}+b{e}^{bt}B=a{A}_0{e}^{(b-a)t}$
Now L.H.S. is equal to $\frac{d(B{e}^{bt})}{dt}$
Hence,
$\begin{array}{}\text{(k is some constant)}& B{e}^{bt}=a{A}_0\int e^{(b-a)t}dt=a{A}_0\frac{e^{(b-a)t}}{(b-a)}+k\end{array}$
which gives $B=a{A}_0\frac{e^{-at}}{(b-a)}+k{e}^{-bt}$
Now, if we add three ODE's, we have,
$\begin{array}{}\text{(}{c}_0\text{ is some constant)}& \frac{d(A+B+C)}{dt}=0⟹A+B+C=c_0\end{array}$
Now put $A$,$B$ to get $C$.

If you solve the first two equations as Avatar suggested you don't really need to substitute into $(3)$ to find $C$ since $(1)+(2)+(3)$ gives
$\frac{d}{dt}(A+B+C)=\frac{dA}{dt}+\frac{dB}{dt}+\frac{dC}{dt}=0⟺A+B+C=\text{const}⟺C=\text{const}-A-B.$