Find all power series solutions of the form $W=\sum_{n=0}^{\infty} a_{n} x^{n}$ to the equation

$$-W^{\prime \prime}+2 x W^{\prime}=E W,$$

for $E$ a real constant.

Impose the condition $W(0)=0$ and determine those values of $E$ for which your power series gives polynomial solutions (i.e., $a_{n}=0$ for $n$ sufficiently large). Give the values of $E$ for which the corresponding polynomials have degree less than 6 , and compute these polynomials.

Hence, or otherwise, find a polynomial solution of

$$-W^{\prime \prime}+2 x W^{\prime}=x-\frac{4}{3} x^{3}+\frac{4}{15} x^{5},$$

satisfying $W(0)=0$.