Homogenous analytic solution to simultaneous differential equations

I am looking for an analytical closed form homogenous solution for the attached simultaneous differential equations. The equations are equations of motion and electrostatic potential. c's and e's are arbitrarily constants representing elastic and piezoelectric coupling. There are solutions for specific cases and there is a solution for an isotropic elastic material in which c's are replaced by Lame constants and Helmholtz decomposition is used. Attached is the solution for the case of transversely isotropic piezoelectric cylinder. Solutions for the case of axisymmetric loading of anisotropic material are available. The following tasks will help in finishing the project: Task 1: Write the equations of the attached paper in terms of r. Task 2: Review my provided equations. (Are they correct?) Task 3: Use Helmholtz decomposition to decompose my equations and write them in terms of r. Task 4: Using the attached paper a similar solution can be provided If you can provide solution for the case of isotropic elastic and radially polarized piezoelectric material then it would be great.