I have an ODE of the type

dy/dt = c1*x1 + c2*x2

(x1,x2=variables, c1,c2=constants)

i would like to solve it numerically using an implicit euler.
the euler backward algorithm is

I have two problems:
first of all, the derivative is not depending on y - just on other variables and secondly: i dont know the future inputs of x1 and x2. so how may i solve this? i know x1 and x2 at every present (and past) step. I read something about newton algorithm but I didn't really get it. i am capable of solving this using explicit algorithms, but that i dont wanna do (for stability reasons)