 Solution

The difference of two solutions would satisfy $Ly=0$ with $y(0)=0$, $y^{\prime }(1)=0$. This would be a zero eigenvalue of the homogeneous problem, but part (b) shows there is none. Hence at most one solution exists. For continuous forcing the standard second-order existence construction also works: the free initial slope is fixed by an equation with coefficient $\cos 1\ne 0$. Thus for the stated regular boundary-value problem the solution is unique even with nonzero endpoint data.