Maybe you mean that y(0)=1 and y'(0)=6. Because otherwise this is not an initial value problem. (Rather what you are posing is a Sturm-Louiville problem and all sort of strange stuff can go on there. )
Find a member of the family of sol'ns of ty'' - y' = 0 satisfying the boundary conditions y(0) = 1, y'(1) = 6. Does the theorem below guarantee that this sol'n is unique?
THEOREM: Existence of a unique sol'n
(Let a_n(t), a_(n-1)(T), ..., a_(1)(t), a_(0)(t) and g(t) be continuous on an interval I and let a_n(t) not equal 0 for every t in this interval. If t = t_0 is any point in the interval, then a sol'n of the IVP (non-homogeneous eq.) exists on the interval and is unique.)
Maybe you mean that y(0)=1 and y'(0)=6. Because otherwise this is not an initial value problem. (Rather what you are posing is a Sturm-Louiville problem and all sort of strange stuff can go on there. )