The present paper deals with an eigenvalue problem which describes an everted neo-hookean spherical shell which its outer surface is deformed in compression under hydrostatic pressure. Our approach is based on mathematical modeling using a differential equation of order four and boundary conditions including two differential equations of order two and three. We solve the above mentioned problem using two different expansions of WKB method. We also investigate how to apply the numerical compound matrix on the problem and show the application of Runge-Kutta-Fehlberg and Newton-Raphson numerical algorithm. Finally, by comparing the data obtained from these two methods (numerical and WKB), we not only learn about the turning point, we also find out that the reason of the difference between the results of the two methods is this turning point.