Prove Green's identity (int_{a}^{b}(u mathcal{L} v-v mathcal{L} u) d x=left.left[pleft(u v^{prime}-v u^{prime} ight) ight] ight|_{a} ^{b}) for the general Sturm-Liouville operator (mathcal{L}).

Prove Green's identity \(\int_{a}^{b}(u \mathcal{L} v-v \mathcal{L} u) d x=\left.\left[p\left(u v^{\prime}-v u^{\prime}\right)\right]\right|_{a} ^{b}\) for the general Sturm-Liouville operator \(\mathcal{L}\).