d^2y/dx^2 +5dy/dx+6y=3e^(-2x)+e^(3x)

I can't seems to recognize the particular solution

2. $
y_p (x) = Axe^{ - 2x} + Be^{3x}
$

you might also read the following article:

Method of undetermined coefficients - Wikipedia, the free encyclopedia

in particular the following paragraph:

f a term in the above particular solution for y appears in the homogeneous solution, it is necessary to multiply by a sufficiently large power of x in order to make the two solutions linearly independent. If the function of x is a sum of terms in the above table, the particular solution can be guessed using a sum of the corresponding terms for y.