#LyX 2.3 created this file. For more info see http://www.lyx.org/ \lyxformat 544 \begin_document \begin_header \save_transient_properties true \origin unavailable \textclass book \use_default_options true \maintain_unincluded_children false \language spanish \language_package default \inputencoding auto \fontencoding global \font_roman "default" "default" \font_sans "default" "default" \font_typewriter "default" "default" \font_math "auto" "auto" \font_default_family default \use_non_tex_fonts false \font_sc false \font_osf false \font_sf_scale 100 100 \font_tt_scale 100 100 \use_microtype false \use_dash_ligatures true \graphics default \default_output_format default \output_sync 0 \bibtex_command default \index_command default \paperfontsize default \spacing single \use_hyperref false \papersize default \use_geometry false \use_package amsmath 1 \use_package amssymb 1 \use_package cancel 1 \use_package esint 1 \use_package mathdots 1 \use_package mathtools 1 \use_package mhchem 1 \use_package stackrel 1 \use_package stmaryrd 1 \use_package undertilde 1 \cite_engine basic \cite_engine_type default \biblio_style plain \use_bibtopic false \use_indices false \paperorientation portrait \suppress_date false \justification true \use_refstyle 1 \use_minted 0 \index Index \shortcut idx \color #008000 \end_index \secnumdepth 3 \tocdepth 3 \paragraph_separation indent \paragraph_indentation default \is_math_indent 0 \math_numbering_side default \quotes_style french \dynamic_quotes 0 \papercolumns 1 \papersides 1 \paperpagestyle default \tracking_changes false \output_changes false \html_math_output 0 \html_css_as_file 0 \html_be_strict false \end_header \begin_body \begin_layout Standard Un \series bold problema de valores iniciales \series default real es uno de la forma \begin_inset Formula \[ \left\{ \begin{aligned}\dot{x} & =f(t,x),\\ x(t_{0}) & =x_{0}, \end{aligned} \right. \] \end_inset dado por \begin_inset Formula $f:\Omega\subseteq\mathbb{R}\times\mathbb{R}^{n}\to\mathbb{R}^{n}$ \end_inset con \begin_inset Formula $\Omega$ \end_inset abierto y \begin_inset Formula $(t_{0},x_{0})\in\Omega$ \end_inset , y donde \begin_inset Formula $x:I\subseteq\mathbb{R}\to\mathbb{R}^{n}$ \end_inset es la incógnita, siendo \begin_inset Formula $I$ \end_inset un entorno de \begin_inset Formula $t_{0}$ \end_inset . \end_layout \begin_layout Standard El problema está \series bold bien planteado \series default en un intervalo \begin_inset Formula $[a,b]\subseteq I$ \end_inset si tiene solución única en \begin_inset Formula $[a,b]$ \end_inset y para todo \begin_inset Formula $\varepsilon>0$ \end_inset existe un \begin_inset Formula $\delta>0$ \end_inset tal que si \begin_inset Formula $c\in(-\varepsilon,\varepsilon)$ \end_inset y \begin_inset Formula $e:[a,b]\to\mathbb{R}^{n}$ \end_inset es tal que \begin_inset Formula $|e(t)|<\varepsilon$ \end_inset para todo \begin_inset Formula $t\in[a,b]$ \end_inset , entonces el \series bold problema perturbado \series default \begin_inset Formula \[ \left\{ \begin{aligned}\dot{z} & =f(t,z)+e(t),\\ z(t_{0}) & =x_{0}+c, \end{aligned} \right. \] \end_inset tiene solución única. \end_layout \begin_layout Standard Como \series bold teorema \series default , si \begin_inset Formula $D\coloneqq [a,b]\times\mathbb{R}$ \end_inset , \begin_inset Formula $t_{0}\in[a,b]$ \end_inset y \begin_inset Formula $f:D\to\mathbb{R}$ \end_inset es continua y lipschitziana en la segunda variable en todo \begin_inset Formula $D$ \end_inset , entonces \begin_inset Formula \[ \left\{ \begin{aligned}\dot{x} & =f(t,x),\\ x(t_{0}) & =x_{0} \end{aligned} \right. \] \end_inset está bien planteado. \end_layout \begin_layout Standard En adelante supondremos que el dominio de \begin_inset Formula $x$ \end_inset incluye un intervalo \begin_inset Formula $[a,b]$ \end_inset y \begin_inset Formula $t_{0}=a$ \end_inset . No siempre se puede resolver un problema de valores iniciales de forma analítica, por lo que usamos métodos de resolución numérica, que aproximan \begin_inset Formula $x|_{[a,b]}$ \end_inset creando una partición \begin_inset Formula $a=t_{0}<\dots