Question: Fourth-order ODE problem

Whoever is in charge of the question post management, please do not delete my question again without even notifying me. I have deleted the original post (which was a follow-up question to an old thread) so that my question can be viewed by others as a fresh post. Thank you!

Now to the question:

I am trying to solve a fourth-order ODE. Two of the boundary conditions are y(1)=0 and diff(y(x),x)=0 at x=1. The other two boundary conditions are y(x) and diff(y(x),x) both being finite at x=0. I have set the problem up as follows:

restart:
assume(n::posint):
additionally(n>3):
dsolve({diff(y(x),x$4)+(2/x)*diff(y(x),x$3)-((2*(n-1)^2+1)/(x^2))*diff(y(x),x$2)+((2*(n-1)^2+1)/(x^3))*diff(y(x),x)+(((n-1)^4-4*(n-1)^2)/(x^4))*y(x)=8*x^(n-1)*(n-n*x^2-x^2)+(n+1)*(n+3)*(n-3)*(n-5)/12-(n+2)*(n+4)*(n-4)*(n-6)*x/12,y(1)=0,eval(diff(y(x),x),x=1)=0,eval(diff(y(x),x),x=0)=eval(diff(y(x),x),x=0)},y(x)) assuming eval(diff(y(x),x),x=0)::finite;

The code returned an answer that unexpectedly still contained an unresolved constant (i.e. _C5) inside the coefficient in front of x^(-n+3). The reason why I set n-3 to be a positive integer was that I intended Maple to equate the coefficient in front of x^(-n+3) to 0 and solve for _C5. This way the finite value boundary conditions could be satisfied. For some reason, Maple did not pick up this assumption. Also, I did not include "assuming y(0)::finite" in the code because Maple complained about imposing assumptions on a constant.

Does anyone have any suggestions for me? Thank you for your insight.