Nonhomogeneous Help on guessing

hello can someone explain how to guess the yp for this Non-homogeneous differential equation

y'' - 2y' + y = te^t

characteristic polynomial: (y - 1)^2 so the characteristic roots are: y1=y2= 1

c1 and c2 are constant

for yh = (c1)e^t + (c2)te^t

please explained how to guess for te^t

HallsofIvy
Any time you have a polynomial or power of t on the right side, try the polynomial up to the highest power. For t you would try At+ B. Here you have $te^t$ so try $(At+ B)e^t$.
Here, both $e^t$ and $te^t$ are solutions to the homogeneous solution. Multiplying $(At+ B)e^t$ by t would give us $(At^2+ Bt)e^t$ but since that still contains $te^t$ which is a solution to homogeneous equation, we multiply by t again: try $(At^3+ Bt^2)e^t$.