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First I found the general solution which came out to be, Ae^3x + Be^-2x

I then Substituted y=ae^-x + bx + c

y'=-ae^-x + b

y"=ae^-x

Then I just compared the coefficients to get a=-1/4, B=-2 and C=-1/6

So I am getting y = Ae^3x + Be^-2x -(e^-x)/4 - 2x - 1/6

Im not sure if this is right, I have done the rest but I get some funny answers for A and B so I was wondering if someone could verify if this answer is right. Thanks