  In general description of the Raychaudhuri equation it is found that this
first order non-linear differential equation can be written as a second order
linear differential equation in the form of Harmonic Oscillator with varying
frequency. Further, the integrability of the Raychaudhuri equation has been
studied and also the expansion scalar is obtained in an explicit form.
Subsequently, $f(R)$ gravity theory has been studied in the background of
inhomogeneous FLRW spacetime with an aim to formulate the Raychaudhuri
Equation. A congruence of time-like geodesics has been investigated using the
Raychaudhuri Equation to examine whether the geodesics converge or not and some
possible conditions are determined to avoid singularity. Finally, a brief
quantum description has been presented.

General Relativity and Quantum Cosmology