Chapter 13, Problem 13RE

To determine

To show: The functions $u_m\left(x\right)=Y_0\left({\alpha }_{m}a\right)J_0\left({\alpha }_{m}x\right)-J_0\left({\alpha }_{m}a\right)Y_0\left({\alpha }_{m}x\right)$ and $u_n\left(x\right)=Y_0\left({\alpha }_{n}a\right)J_0\left({\alpha }_{n}x\right)-J_0\left({\alpha }_{n}a\right)Y_0\left({\alpha }_{n}x\right)$ are orthogonal with respect to the weight function $p\left(x\right)=x$ on the interval $\left[a,b\right]$.