This is how i worked it out so far:

Using trial solution $\displaystyle x = At^{2}e^{-t}$

$\displaystyle \frac{dx}{dt} = 0$

$\displaystyle \frac{d^{2}x}{dt^{2}} = 0$

Mr F says: I have no idea how you got the above results but they are wrong.
therefore $\displaystyle At^{2}e^{-t} = e^{-t}$

comparing coefficients:

$\displaystyle At^{2} = 1$

$\displaystyle t = \sqrt{\frac{1}{A}} $ ....Particular Integral

Mr F says: None of the above is correct. And there are various careless errors in your complementary solution as well: should be $\displaystyle x = ({\color{red}a} + Bt) e^{-{\color{red}t}}$. The pronumeral A should not be used in the complementary solution because this symbol is already used in the question to mean something else.
[snip]