  This paper provides a pedagogical introduction to the classical nonlinear
stability analysis of the plane Poiseuille and Couette flows. The whole
procedure is kept as simple as possible by presenting all the logical steps
involved in the application of the energy method and leading to the
Euler-Lagrange equations. Then, the eigenvalue problems needed for the
evaluation of the nonlinear energy threshold of the Reynolds number for
stability are formulated for transverse modes and for longitudinal modes. Such
formulations involve the streamfunction and, in the case of longitudinal modes,
also the streamwise component of velocity. An accurate numerical solution of
the eigenvalue problems, based on Galerkin's method of weighted residuals with
the test functions expressed in terms of Chebyshev polynomials, is discussed in
details. The numerical codes developed for the software Mathematica 14
(\copyright{} Wolfram Research, Inc.) are also presented. A critical analysis
of the obtained results is finally proposed.
