A Sturm-Liouville's problem is often presented with the following boundary values: \begin{align} \alpha_1\,y(a) + \beta_1\, y'(a) &= 0 \\ \alpha_2\,y(b) + \beta_2\, y'(b) &= 0 \end{align}
Would it have sense to use higher derivatives ? for instance: \begin{align} \alpha_1\,y(a) + \beta_1\, y'(a) + \gamma_1\, y''(a) &= 0 \\ \alpha_2\,y(b) + \beta_2\, y'(b) + \gamma_2\, y''(b) &= 0 \end{align} and in such a case, should we expect to have a solution to the BVP ?