Prove that if \(u(x)\) and \(v(x)\) satisfy the general homogeneous boundary conditions

\[\begin{array}{r} \alpha_{1} u(a)+\beta_{1} u^{\prime}(a)=0 \\ \alpha_{2} u(b)+\beta_{2} u^{\prime}(b)=0 \tag{6.173} \end{array}\]

at \(x=a\) and \(x=b\), then

\[p(x)\left[u(x) v^{\prime}(x)-v(x) u^{\prime}(x)\right]_{x=a}^{x=b}=0\]