Summary: Exact solutions for a class of non-linear second-order differential equations arising in a third grade fluid flow, at a rotating cylinder (unbounded domain case) and between rotating cylinders (bounded domain case), are obtained. Furthermore, the exact solutions are compared with the numerical ones. It is observed that the difference between the exact and the numerical solutions is about 1% for small \(R\) (the non-dimensional distance between the cylinders) and is about 3% when \(R=100\). This difference increases with an increasing \(R\). Moreover, for large \(R\) it is not easy to obtain meaningful results numerically. Hence, these exact solutions for various values of the parameters \(R\) and \(\omega\) (rotation parameter) are useful for experimental and numerical studies, and warrant further study.