Particular solution of $y''−2y'+y=e^t+t$

Question

how to guess a particular solution for the following DV?

$y''−2y'+y=e^t+t$

Answer 1

Note that both $e^t$ and $te^t$ are solutions to the homogeneous equation $$y′′−2y′+y= 0$$

Thus for your particular solution you need to choose $$ y_p = At^2e^t+Bt+C$$ and find the constants by plugging $y_p$ in $$ y′′−2y′+y=e^t+t$$