Finding particular solution of $y'' + ay' + by = At^{2}e^{-kt^{2}}$

Question

If possible, can someone assist me in finding the particle (non-homogeneous) solution for the general differential equation $$y'' + ay' + b y = At^{2}e^{-kt^{2}}$$ Where $a $, $b $, $A$ and $k$ are positive constants; also $y $ is only dependent on $t $. I took calculus a decade ago but remember that to use undetermined coefficients approach, the driving function must have a finite derivative family; and this one does not.

Answer 1

Undetermined coefficient only work on certain types of function (as you note). $e^{-kt^2}$ is not of that type. So you cannot use undetermined coefficients.

The other common method to find a particular solution (after finding the general solution of the homogeneous equation) is "variation of parameters". This method should be in differential equations textbooks.