Solve the following boundary value problem $$y''+\left(\frac{(1-e^{-4x})^2}{2x^3}-\frac{2(1-e^{-4x})}{x}\right)y'-\frac{(1-e^{-4x})^4}{16x^4} y=0, \quad y(0)=y(\tfrac{1}{2})=0.$$
Note: I attempted to use the generalized exponential integral $$Ein(z)\equiv \int_{0}^{z}\frac{1-e^{-t}}{t}\mathrm{d}t$$ to represent the solution. But it seems that I cannot do this directly.