In this work we will study some methods for the solution of Ordinary Differential Equations
(ODE) of the first order, in order to bring a better compression on them, also we will also
study some numerical methods to determine approximate values of the solution of an ODE,
knowing its initial value. Some applications will be presented, through the numerical solution
of initial value problems with the first-order ODE. In this way, we will study some concepts
about the first-order ODE, where we will analyze the Existence and Oneness Theorem for an
initial value problem (PVI), and also some methods for the solution of the first-order ODE,
such as linear, separable, homogeneous and exact. Due to the difficulty of solving some ODE,
and in order to determine approximate values of the PVI solution, also we will study some
single-step numerical methods, such as the Euler method, the Taylor method of higher-order,
and Runge-Kutta. To exemplify some of the numerical methods that we will study, we bring
some applications considering PVIs.
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