This monograph presents an introduction to Variational Methods, which today form an important
method that is applied in the field of differential equations. Thus, we sought to answer the
following guiding question: how to solve ordinary differential equations (ode) using Variational
Methods? This research has as main objective to determine necessary and sufficient conditions
for certain ordinary differential equations to have a solution via Variational Methods. For this,
initially, a brief review of measure and Lebesgue spaces was made to generalize the concept
of Riemann’s integral; from this, the concept of the weak derivative was defined, followed by
the well-known Sobolev spaces. Thus, in these spaces, what we call a weak solution of the
functional associated with the given equation was established, to later solve the Edo by the
Variational Methods. As for the methodology used in this work, we have exploratory and bibliographical
research, and a qualitative approach. As a result of this study, we highlight the use of
the Mountain Pass Theorem, which provides some functional conditions, including the Palais
Smale condition, under which the functional has a critical point. Thus, Variational Methods are
concerned with finding critical functional points associated with some differential equation.
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