Poly-logarithm as solution of Cauchy-Euler equation, $\zeta (3)$ as initial condition.

Question

A friend is interested in $\zeta (3)$ and particularly in the following question:

Consider the differential equation $$x^2y''-2xy'+2y=-2\zeta(3)+\frac{x^3}{3}\sum _{k=0}^{\infty}\frac{B_kx^{k+2}}{k!(k+2)},$$ where $B_k$ is the $k-$th Bernoulli number. Using the identity $$2\int _0^{s/2}t^2coth(t)dt=\frac{s^3}{12}+\frac{s^2}{2}Li_1\left (e^{-s}\right )-s Li_{2}\left (e^{-s}\right )-Li_{3}\left (e^{-s}\right )+\zeta (3),$$ he is able to find one particular solution, namely, $y=Li_3(e^{-x}),$ and knowing that the homogeneous equation looks like $c_1x^2+c_2x,$ he is able to find the solution $$y(x)=Li_3(e^{-x})+c_1x^2+c_2x,$$ having the property that $y(0)=\zeta (3).$

Question: Is there any initial value conditions that give any interesting solutions with regard to the property of having $\zeta (3)$ as $y(0)$? There is some Sturm-Liouville problem like basis in which one can express this solution?

References of this kind of stuff are welcome.