Chapter 14.2, Problem 18E

(a)

To determine

To show: The separated differential equation for the partial differential equation $a^2\left(\frac{{\partial }^{2}u}{\partial r^2}+\frac{1}{r}\frac{\partial u}{\partial r}+\frac{1}{r^2}\frac{{\partial }^{2}u}{\partial {\theta }^2}\right)=\frac{{\partial }^{2}u}{\partial t^2}$ is $T^{″}+a^2\lambda T=0$ and $r^2{R}^{″}+r{R}^{\prime }+\left(\lambda r^2-\nu \right)R=0$.

(b)

To determine

The solution of separated differential equations obtained in sub-part (a).

(c)

To determine

The Eigenvalues and the Eigen functions for which the non-trivial solution exist.

(d)

To determine

The multiple series solution for the differential equation $a^2\left(\frac{{\partial }^{2}u}{\partial r^2}+\frac{1}{r}\frac{\partial u}{\partial r}+\frac{1}{r^2}\frac{{\partial }^{2}u}{\partial {\theta }^2}\right)=\frac{{\partial }^{2}u}{\partial t^2}$.