1(b)

Use the Exponential Shift Theorem with $$y = u {\rm e}^{2 x}$$ to get $$((D+2)^2 - 4(D+2) + 4) u = D^2 u = x^{-3/2}$$. Integrate twice to get $$u = -4 x^{1/2} + C_1 x + C_2$$. The general solution is $y = (-4 \sqrt{x} + c_1 + c_2 x) {\rm e}^{2 x}$.