The Dynamic Response of a single pile following Leipholz’s rod has been solved. The analysis attempts to provide the solution of the pile if the model of the system is taken to have a non-self-adjoint follower load under the influence of a heterogeneous Winkler soil model. The method of Laplace transform is used to deal with the non-conservative initial-boundary-value problems. In this paper, a close-form solution has been developed to deal with the concave singularity function of the soil subgrade modulus if the admissible function for the pile is satisfied when upset by a time-dependent lateral force. This research domain is still active as research engineers and mathematicians are engaged in developing a deeper understanding of dealing with singularity. The result obtained is compared with those given by static (finite element) analysis.