You lost some stuff in your final answer. I get (unfortunately latex isn't working properly but)
y = \dfrac{c2(x+1)+c1}{c2x + c1}e^{-x}
If c2 not = 0 then divide by c2 and call c1/c2 = c, a constant.
Im all good up until part (d).
In part (c) we have a DE with constant coefficients with = 1.
The general solution for w is
w(x) = C_1e^x + C_2xe^x
w'(x) = C_1e^x + C_2e^x + C_2xe^x
Substituting the formulas for w(x) and w'(x) gives:
y(x) = e^-x * (C_1e^x + C_2e^x + C_2xe^x) / (C_1e^x + C_2xe^x)
Simplifying this down gives:
y(x) = e^-x ( 1 + C_2/(C_1 + C_2x) )
Now how do I go about combining the two constants to only get one in my final solution? I should only get one since the original DE is only first order.