What is the maximum number of linearly independent solutions of the diff. eqn. d4y/dx4=0 with y(0)=1?
And what is the maximum number when the condition given is y'(0)=0?
More abstractly, an nth order linear differential equation will have n independent solutions (its solution space has dimension n) and adding one constraint will reduce that by 1.
Less abstractly, it should be very easy to find the general solution just by integrating four times, then apply the conditions and see what you get!