Hey would you use the Method of Undetermined Coefficients (Method of Good Guessing) to find a particular sol'n to this DE
From the homogenous solution to the differential you find that solutions are:
.
However, when you go to solve for the two particular solutions to the differential, you notice that you already have an exponential to the 2x as a solution of your homogenous. So you need to multiple your original "guess" by , where k is the smallest exponent needed to get a unique guess that is not a solution to our homogenous differential.
Therefore I would solve my particular as:
Now substituting into your original d.e.:
Terms cancel out leaving:
Therefore our particular solution for our exponential is