solve that nasty 4th order polynomial equation you now have, for u. (it's not pretty)
Note that u is just a constant. (up to 4 constants actually)
Integrate u over x to obtain y(x) = u x + const. This is actually up to 4 separate solutions as there are up to 4 distinct roots of your polynomial.
The differential equation is satisfied for all the u's.
did you maybe mean d4y/dx4 + B d3y/dx3 + C = 0 ?