Self-Adjoint Operators

Question

If an Operator $L$ is defined as $Lu=u''$ and $a_1u(0)+b_1u'(0)+c_1u(1)+d_1u'(1)=0$ along with $a_2u(0)+b_2u'(0)+c_2u(1)+d_2u'(1)=0$, then for what values of $a_1,b_1,c_1,etc$ is the operator self-adjoint? Obviously, I see that the operator L is symmetric after expanding it out via integration by parts. Then, I need to assert that $[u'v-uv']$ evaluated at $x=0$ to $x=1$ (boundary upon which this is defined) is equivalent to zero. Any idea how to go about this? I feel like I don't have enough equations to solve for those constants....