I'm trying to find the general solution using reduction of order for this ode
$$y′′ + 2y′ + y = e^{-x}$$
i have found the complementary soultion :
$$c_1e^{-x}+c_2e^{-x}x$$
but im unsure how to use reduction of order method to find the paticular soultion
Since $e^{-t}$ satisfies $$y′′ + 2y′ + y = 0,$$ a particular solution should be of the form $$y_p=at^2e^{-t}$$ which is easy to get $a=\frac12$. Then the GS is $$ y=(c_1+c_2t)e^{-t}+\frac12t^2e^{-t}. $$