The mechanism of instability in fluid mechanics is one of the topics, which has not been fully understood yet, since there are numerous external and internal agents for a flow field to lose its laminar behavior and to find itself in transition to turbulence. In this study, only the effects of pressure gradients are considered. Commonly used methods by many researchers, in this field, try to connect experimental evidence to numerically obtained results. One of the ways to do this matching is linearized (N-S) equations under small disturbances which finally yield the so-called Orr-Sommerfeld equation. In this paper, Orr-Sommerfeld equation is solved to eight-digit accuracy and compared with literature, by using Chebyshev polynomials. After having this accuracy, the flow field has been examined for various pressure gradients. Increasing the pressure gradients either positive or negative, increases the instability region.