So I have this system of differential equations, which I've been trying to solve for quite a while. $$c_1' = a_2c_3-a_3c_2+b_1\\ c_2'=a_3c_1-a_1c_3+b_2\\ c_3'=\lambda(a_1c_2-a_2c_1)+b_3$$ where $a_1, a_2, b_1, b_2, c_1, c_2 \text{ and } c_3$ are functions of $t; \lambda, a_3\text{ and } b_3$ are constants. Given that $a_1, a_2, b_1, b_2$ are known functions, how do I solve for $c_1, c_2 \text{ and } c_3$?
My main goal is to get a differential equation with only one unknown function. I have tried solving them by using matrices and eigenvectors but it doesn't work. I tried solving for one function and substituting back and again it doesn't work. The problem seems to be that $c_1, c_2, c_3$ are missing in $c_1', c_2', c_3'$ equations respectively.