The problem of finite-time and fixed-time stability analysis is considered for a class of nonlinear systems dx/dt = H(x)b(x), where H is homogeneous of a nonnegative degree and b is bounded. It is shown that under certain conditions on b the asymptotic stability of the system implies its finite-time stability or nearly fixed-time stability. The so-called homogeneous extensions are utilized for the analysis. The results are generalized to a class of systems. An example of a mechanical system is presented in the last section to illustrate the obtained results.