Second order ODE with constant coeffiecient? Is the involved method for finding the solution to homogenous ODE required, or is there a shortcut, given the lack of functions of x?
$$\ y'' + ay'=b$$

Considering the problem $$\ y'' + ay'=b$$ you can easily go back to much simpler solutions since $a$ and $b$ are constants.
Rewrite the equation as $$y''+a(y'-\frac ba)=0$$ and now define $z=y'-\frac ba$ which makes $$z'+a z=0$$ the simplest first order differentiatial equation. Solve it for $z$ $$z=c_1 \,e^{-a x}$$ and replace $$y'-\frac ba=c_1 \,e^{-a x}$$ $$y'=\frac ba+c_1 \,e^{-a x}$$ and just integrate one more to get the final solution..
