That's what I would do for starters. After all, you're only concerned with terms up to degree 6.
Start with the assumption that y = a0 + a1t + a2t2 and so on up to degree 8 or so (to get degree 6 term in your second derivative. Calculate y' and y'' and multiply y' by t, then add them together and group them by like powers of t. On the right side, you'll have the power series for e-2t, which looks like 1 - 2t + (2t)2/2! + a few more terms to get to degree 6. Then equate both sides and solve for your coefficients.