let dy/dx = u

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 d^{4}y/dx^{4}+ B d^{3}y/dx^{3}+ C = 0 ?