Transformada de Laplace

<body topmargin=0 leftmargin=0 marginheight=0 marginwidth=0> <table cellpadding=0 cellspacing=0 border=0><tr> <td valign="top" colspan=3 height=24 BGCOLOR="#003366" TEXT="#F7F7FF"> <div> <FONT SIZE="4" COLOR="#FFFFFF" FACE="Bell MT" ><B>| </B><A HREF="index.htm" TARGET="_top" TITLE="Ecuaciones diferenciales"><U><B>Ecuaciones diferenciales</B></U></A><B>  | </B></FONT><A HREF="index.htm" TARGET="_top" TITLE="Ecuaciones diferenciales"><U><B>Home</B></U></A><B> | </B><A HREF="id118.htm" TARGET="_top" TITLE="C o n t e n i d o"><U><B>Contenido</B></U></A><B> | </B><A HREF="id223.htm" TARGET="_top" TITLE="Apuntes"><U><B>Apuntes</B></U></A></div> <div> <A HREF="id226.htm" TARGET="_top" TITLE="Introducción a las ecuaciones diferencia"><U>Introducción a las ecuaciones diferenciales</U></A>   |   <A HREF="id227.htm" TARGET="_top" TITLE="Ecuaciones separables"><U>Ecuaciones separables</U></A>   |   <A HREF="id225.htm" TARGET="_top" TITLE="Ecuaciones lineales de primer orden"><U>Ecuaciones lineales de primer orden</U></A>   |   <A HREF="id228.htm" TARGET="_top" TITLE="Ecuaciones de Bernoulli"><U>Ecuaciones de Bernoulli</U></A>   |   <A HREF="id273.htm" TARGET="_top" TITLE="Ecuaciones de Riccati"><U>Ecuaciones de Riccati</U></A>   |   <A HREF="id229.htm" TARGET="_top" TITLE="Ecuaciones exactas"><U>Ecuaciones exactas</U></A>   |   <A HREF="id230.htm" TARGET="_top" TITLE="Ecuaciones homogéneas"><U>Ecuaciones homogéneas</U></A>   |   <A HREF="id231.htm" TARGET="_top" TITLE="Introducción a las ecuaciones diferencia"><U>Introducción a las ecuaciones diferenciales lineales de segundo orden</U></A>   |   <B>Transformada de Laplace</B></div> <div> </div> </td> </tr><tr> <td valign="top" rowspan="2" width=165 BGCOLOR="#FFFFFF" TEXT="#080000"> <div> <B> <script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script>  <ins class="adsbygoogle" style="display:inline-block;width:160px;height:600px" data-ad-client="ca-pub-1578463498418701" data-ad-slot="3073775660"></ins> <script> (adsbygoogle = window.adsbygoogle || []).push({}); </script>  </B></div> </td> <td valign="top" height=432 BGCOLOR="#FFFFFF" TEXT="#080000"> <div> <B>T r a n s f o r m a d a   d e   L a p l a c e</B></div> <div>  <A NAME="top"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>Top</div> <div>  <script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script>  <ins class="adsbygoogle" style="display:inline-block;width:728px;height:90px" data-ad-client="ca-pub-1578463498418701" data-ad-slot="5388515922"></ins> <script> (adsbygoogle = window.adsbygoogle || []).push({}); </script>  </div> <bR> <div> <A HREF="#definición" TITLE="Transformada de Laplace"><U><IMG BORDER="0" SRC="Symb075c00b7011899660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Definición de la transformada de Laplac</U></A>e</div> <div> <A HREF="#existencia" TITLE="Transformada de Laplace"><U><IMG BORDER="0" SRC="Symb075c00b7011899660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Exixtencia de la transformada de Laplace</U></A>:</div> <div> <A HREF="#teoremaexistenacia" TITLE="Transformada de Laplace"><U><IMG BORDER="0" SRC="Wing0a7300a8011899660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Teorema. Condiciones suficientes para la existencia de la transforma</U></A><A HREF="#teoremaexistenacia" TITLE="Transformada de Laplace"><U>da</U></A></div> <div> <A HREF="#corolario" TITLE="Transformada de Laplace"><U><IMG BORDER="0" SRC="Wing0a7300a8011899660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Corolario</U></A></div> <div> <IMG BORDER="0" SRC="Symb075c00b7011899660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Ejercicios resueltos:</div> <div> <A HREF="#ejercicio_1" TITLE="Transformada de Laplace"><U><B><IMG BORDER="0" SRC="Wing0a7300a8011899660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0"><IMG SRC="fea1f170.gif" border=0 width="287" height="23" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A></div> <div> <TABLE BORDER="0" CELLPADDING="0" CELLSPACING="1" WIDTH="342" BORDERCOLORLIGHT="#FFFFFF" BORDERCOLORDARK="#FFFFFF" FRAME="VOID" RULES="NONE" ALIGN="RIGHT" HSPACE="0" VSPACE="0" BGCOLOR=#FFFFFF > <tR> <tD VALIGN=TOP BGCOLOR=#FFFFFF HEIGHT= 288 ><NOBR><div>  <script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script>  <ins class="adsbygoogle" style="display:inline-block;width:336px;height:280px" data-ad-client="ca-pub-1578463498418701" data-ad-slot="7754199516"></ins> <script> (adsbygoogle = window.adsbygoogle || []).push({}); </script>  </div> </NOBR></tD> </tR> </TABLE></div> <div> <A HREF="#ejercicio2" TITLE="Transformada de Laplace"><U><B><IMG BORDER="0" SRC="Wing0a7300a8011899660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0"><IMG SRC="100e34180.gif" border=0 width="308" height="24" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A></div> <div> <A HREF="#ejercicio_3" TITLE="Transformada de Laplace"><U><B><IMG BORDER="0" SRC="Wing0a7300a8011899660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0"><IMG SRC="100f3d1c0.gif" border=0 width="317" height="28" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A></div> <div> <A HREF="#ejercicio_4" TITLE="Transformada de Laplace"><U><B><IMG BORDER="0" SRC="Wing0a7300a8011899660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0"><IMG SRC="10163d1c0.gif" border=0 width="317" height="28" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A></div> <div> <A HREF="#ejercicio_5" TITLE="Transformada de Laplace"><U><B><IMG BORDER="0" SRC="Wing0a7300a8011899660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0"><IMG SRC="ff1451c0.gif" border=0 width="325" height="28" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A></div> <div> <A HREF="#ejercicio_6" TITLE="Transformada de Laplace"><U><B><IMG BORDER="0" SRC="Wing0a7300a8011899660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0"><IMG SRC="101b4d170.gif" border=0 width="333" height="23" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A></div> <div> <A HREF="#ejercicio_7" TITLE="Transformada de Laplace"><U><B><IMG BORDER="0" SRC="Wing0a7300a8011899660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0"><IMG SRC="101c4f170.gif" border=0 width="335" height="23" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A></div> <div> <A HREF="#ejercicio_8" TITLE="Transformada de Laplace"><U><B><IMG BORDER="0" SRC="Wing0a7300a8011899660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0"><IMG SRC="10105c180.gif" border=0 width="348" height="24" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A></div> <div> <A HREF="#ejercicio_9" TITLE="Transformada de Laplace"><U><B><IMG BORDER="0" SRC="Wing0a7300a8011899660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0"><IMG SRC="10115b180.gif" border=0 width="347" height="24" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A></div> <div> <A HREF="#unicidad_de_la_transformada" TITLE="Transformada de Laplace"><U><IMG BORDER="0" SRC="Symb075c00b7011899660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Unicidad de la transformada de Laplace</U></A></div> <div> <IMG BORDER="0" SRC="Symb075c00b7011899660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Propiedades de la Transformada de Laplace</div> <div> <A HREF="#linealidad" TITLE="Transformada de Laplace"><U>Linealidad de la transformada</U></A></div> <div> <A HREF="#tabla_de_transformadas_de_laplace" TITLE="Transformada de Laplace"><U><IMG BORDER="0" SRC="Symb075c00b7011899660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Tabla de transformadas de Laplace</U></A></div> <div> <A HREF="#transformada_inversa_de_laplace" TITLE="Transformada de Laplace"><U><IMG BORDER="0" SRC="Symb075c00b7011899660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Transformada de Laplace inversa</U></A> </div> <div> <A HREF="#linealidad_de_la_transformada_de" TITLE="Transformada de Laplace"><U>Linealidad de la transformadas de Laplace inversa</U></A></div> <div> <A HREF="#transformadas_de_derivadas" TITLE="Transformada de Laplace"><U><IMG BORDER="0" SRC="Symb075c00b7011899660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Transformadas de derivadas</U></A></div> <div> <A HREF="#teorema___transformada_de_una_derivada" TITLE="Transformada de Laplace"><U>Teorema. Transformada de una derivada</U></A></div> <div> <IMG BORDER="0" SRC="Symb075c00b7011899660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Solución de ecuaciones diferenciales mediante la transformada de Laplace</div> <div> <IMG BORDER="0" SRC="Symb075c00b7011899660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Problemas de aplicación</div> <div> <IMG BORDER="0" SRC="Symb075c00b7011899660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Solución de ejercicios enunciados en el libro "Ecuaciones diferenciales", séptima edición, de Dennis G. Zill: <A HREF="id271.htm" TARGET="_top" TITLE="7.1"><U>Ejercicios 7.1</U></A> </div> <div> <A HREF="#apendice" TITLE="Transformada de Laplace"><U><IMG BORDER="0" SRC="Symb075c00b7011899660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Apéndice</U></A>:</div> <div> <IMG BORDER="0" SRC="Wing0a7300a8011899660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Integral impropia</div> <div> <A HREF="#funcion_gamma" TITLE="Transformada de Laplace"><U><IMG BORDER="0" SRC="Wing0a7300a8011899660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">La función gamma</U></A></div> <div> <IMG BORDER="0" SRC="Wing0a7300a8011899660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Fracciones parciales</div> <div> <IMG BORDER="0" SRC="Wing0a7300a8011899660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Ecuaciones simultáneas</div> <div> <IMG BORDER="0" SRC="Wing0a7300a8011899660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">F<A HREF="#continua_por_tramos" TITLE="Transformada de Laplace"><U>unción continua por tramos</U></A></div> <div> <A HREF="#orden_exponencial" TITLE="Transformada de Laplace"><U><IMG BORDER="0" SRC="Wing0a7300a8011899660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Función de orden exponencia</U></A>l</div> <div> <IMG BORDER="0" SRC="Wing0a7300a8011899660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Nota historica</div> <bR> <DIV ALIGN="LEFT"></DIV> <div> <IMG SRC="1009bc050.gif" border=0 width="700" height="5" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="definición"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>D</div> <div> <I><B>Definición. Transformada de Laplace:</B></I></div> <div> <IMG SRC="ffbe50f0.gif" border=0 width="997" height="271" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="existencia"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>Existencia</div> <div> <B>Existencia de la transf</B><B>or</B><B>mada de La</B><B>p</B><B>la</B><B>c</B><B>e:</B></div> <div> Antes de proceder a deducir la transformada de Laplace para una función <I>f</I> en particular hay que estar seguros que para esa clase de funciones existe la transformada. Antes de enunciar el teorema de existencia debemos conocer dos conceptos en los que se sustenta el teorema: <I>Función continua por tramos</I> y <I>función de orden exponencial</I>.</div> <div>  <A NAME="teoremaexistenacia"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>T</div> <div> <I><B>Teorema. Condiciones suficie</B></I><I><B>nt</B></I><I><B>es para la existencia:</B></I><FONT SIZE="4" COLOR="#FFFFFF" FACE="Georgia" >Existenacia</FONT></div> <div> <IMG SRC="ff817670.gif" border=0 width="535" height="103" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="corolario"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>Corolario</div> <div> <I><B>Corolario:</B></I></div> <div> <I><B><IMG SRC="1027313c0.gif" border=0 width="561" height="60" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></I></div> <bR> <div>  <A NAME="unicidad_de_la_transformada"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>Unicidad de la transformada</div> <div> <B>Unicidad de la transformada de Laplace:</B></div> <div> <B><IMG SRC="102641500.gif" border=0 width="833" height="80" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <div> <B>Propiedades operacionales de la Transformada de Laplace:</B></div> <div> No es necesario recurrir cada vez a la <A HREF="#definición" TITLE="Transformada de Laplace"><U>definición</U></A> para hallar la transformada de Laplace de una función <I>f</I>(<I>t</I>) dada, en algunas ocasiones este trabajo innecesario es muy laborioso. Para facilitar las cosas es conveniente conocer y aplicar ciertas propiedades de las transformadas.</div> <div>  <A NAME="linealidad"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>Linealidad</div> <div> <B>Teorema. Propiedad de Linea</B><B>lid</B><B>ad:</B></div> <div> <B><IMG SRC="ffe8f370.gif" border=0 width="911" height="55" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <div> La transformada de Laplace de una combinación lineal de funciones es igual a la combinación lineal de sus transformadas y se extiende a una combinación lineal finita de trnasformadas de Laplace.</div> <bR> <div>  <A NAME="tabla_de_transformadas_de_laplace"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>Tabla de transformadas de Laplace</div> <div> <B>Tabla de transformadas de Laplace:</B></div> <div> <B> <TABLE BORDER="2" CELLPADDING="2" CELLSPACING="1" WIDTH="764" BORDERCOLORLIGHT="#C0C0C0" BORDERCOLORDARK="#808080" FRAME="BOX" RULES="ALL" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP HEIGHT= 25 WIDTH="196"><div> <B>F u n c i ó n</B></div> </tD> <tD VALIGN=TOP WIDTH="346"><div> <B>T r a n s f o r m a d a</B></div> </tD> <tD VALIGN=TOP WIDTH="196"><div> <B>D o m i n i o </B></div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 32 WIDTH="196"><div> <IMG SRC="102c25180.gif" border=0 width="37" height="24" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP WIDTH="346"><div> <IMG SRC="102d28180.gif" border=0 width="40" height="24" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP WIDTH="196"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="32" ALIGN="bottom" WIDTH="196" HSPACE="0" VSPACE="0"></tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 52 WIDTH="196"><div> 1</div> </tD> <tD VALIGN=TOP WIDTH="346"><div> <IMG SRC="102f102c0.gif" border=0 width="16" height="44" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP WIDTH="196"><div> <IMG SRC="103027130.gif" border=0 width="39" height="19" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 55 WIDTH="196"><div> <I>t</I></div> </tD> <tD VALIGN=TOP WIDTH="346"><div> <IMG SRC="1031192f0.gif" border=0 width="25" height="47" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP WIDTH="196"><div> <IMG SRC="103227130.gif" border=0 width="39" height="19" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 53 WIDTH="196"><div> <IMG SRC="103914180.gif" border=0 width="20" height="24" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP WIDTH="346"><div> <IMG SRC="1038792d0.gif" border=0 width="121" height="45" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP WIDTH="196"><div> <IMG SRC="103a6f180.gif" border=0 width="111" height="24" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 56 WIDTH="196"><div> <IMG SRC="103b14180.gif" border=0 width="20" height="24" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP WIDTH="346"><div> <IMG SRC="103c68300.gif" border=0 width="104" height="48" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP WIDTH="196"><div> <IMG SRC="103d701c0.gif" border=0 width="112" height="28" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 48 WIDTH="196"><div> <IMG SRC="103e1b170.gif" border=0 width="27" height="23" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP WIDTH="346"><div> <IMG SRC="103f28280.gif" border=0 width="40" height="40" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP WIDTH="196"><div> <IMG SRC="104027130.gif" border=0 width="39" height="19" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 55 WIDTH="196"><div> <IMG SRC="10413d140.gif" border=0 width="61" height="20" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP WIDTH="346"><div> <IMG SRC="10423c2f0.gif" border=0 width="60" height="47" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP WIDTH="196"><div> <IMG SRC="1043301c0.gif" border=0 width="48" height="28" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 55 WIDTH="196"><div> <IMG SRC="10443d140.gif" border=0 width="61" height="20" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP WIDTH="346"><div> <IMG SRC="10463c2f0.gif" border=0 width="60" height="47" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP WIDTH="196"><div> <IMG SRC="10452c1b0.gif" border=0 width="44" height="27" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 55 WIDTH="196"><div> <IMG SRC="104733140.gif" border=0 width="51" height="20" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP WIDTH="346"><div> <IMG SRC="10483d2f0.gif" border=0 width="61" height="47" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP WIDTH="196"><div> <IMG SRC="104927130.gif" border=0 width="39" height="19" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 55 WIDTH="196"><div> <IMG SRC="104a33140.gif" border=0 width="51" height="20" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP WIDTH="346"><div> <IMG SRC="104b3d2f0.gif" border=0 width="61" height="47" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP WIDTH="196"><div> <IMG SRC="104c27130.gif" border=0 width="39" height="19" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 32 WIDTH="196"><div> <IMG SRC="104d6d180.gif" border=0 width="109" height="24" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP WIDTH="346"><div> <B><IMG SRC="104e75180.gif" border=0 width="117" height="24" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> </tD> <tD VALIGN=TOP WIDTH="196"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="32" ALIGN="bottom" WIDTH="196" HSPACE="0" VSPACE="0"></tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 32 WIDTH="196"><div> <IMG SRC="104f2c180.gif" border=0 width="44" height="24" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP WIDTH="346"><div> <IMG SRC="105064180.gif" border=0 width="100" height="24" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP WIDTH="196"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="32" ALIGN="bottom" WIDTH="196" HSPACE="0" VSPACE="0"></tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 36 WIDTH="196"><div> <IMG SRC="10512f180.gif" border=0 width="47" height="24" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP WIDTH="346"><div> <IMG SRC="1052a91c0.gif" border=0 width="169" height="28" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP WIDTH="196"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="36" ALIGN="bottom" WIDTH="196" HSPACE="0" VSPACE="0"></tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 37 WIDTH="196"><div> <IMG SRC="1053391d0.gif" border=0 width="57" height="29" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP><NOBR><div> <IMG SRC="10545f1d0.gif" border=0 width="351" height="29" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> <tD VALIGN=TOP WIDTH="196"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="37" ALIGN="bottom" WIDTH="196" HSPACE="0" VSPACE="0"></tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 36 WIDTH="196"><div> <IMG SRC="10553c1c0.gif" border=0 width="60" height="28" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP WIDTH="346"><div> <IMG SRC="105643180.gif" border=0 width="67" height="24" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP WIDTH="196"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="36" ALIGN="bottom" WIDTH="196" HSPACE="0" VSPACE="0"></tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 60 WIDTH="196"><div> <IMG SRC="105731180.gif" border=0 width="49" height="24" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP WIDTH="346"><div> <IMG SRC="10584f340.gif" border=0 width="79" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP WIDTH="196"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="60" ALIGN="bottom" WIDTH="196" HSPACE="0" VSPACE="0"></tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 57 WIDTH="196"><div> <IMG SRC="10594c180.gif" border=0 width="76" height="24" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP WIDTH="346"><div> <IMG SRC="105a65310.gif" border=0 width="101" height="49" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP WIDTH="196"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="57" ALIGN="bottom" WIDTH="196" HSPACE="0" VSPACE="0"></tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 57 WIDTH="196"><div> <IMG SRC="105c49170.gif" border=0 width="73" height="23" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP WIDTH="346"><div> <IMG SRC="105b65310.gif" border=0 width="101" height="49" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP WIDTH="196"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="57" ALIGN="bottom" WIDTH="196" HSPACE="0" VSPACE="0"></tD> </tR> </TABLE></B></div> <div>  <A NAME="transformada_inversa_de_laplace"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>Transformada inversa de Laplace</div> <div> <B>Transformada de Laplace inversa:</B></div> <div> <B><IMG SRC="102381390.gif" border=0 width="897" height="57" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <div>  <A NAME="linealidad_de_la_transformada_de"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>Linealidad de la transformada de Laplace inversa:</div> <div> <B>Linealidad de la transformada de Laplace inversa:</B></div> <div> <B><IMG SRC="1024036f0.gif" border=0 width="1027" height="111" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <div>  <A NAME="transformadas_de_derivadas"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>Transformadas de derivadas:</div> <div> <B>Transformadas de derivadas:</B></div> <div> <B><IMG SRC="1028fdb90.gif" border=0 width="1021" height="441" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <div>  <A NAME="teorema___transformada_de_una_derivada"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>Teorema.  Transformada de una derivada</div> <div> <B>Teorema.  Transformada de una derivada:</B></div> <div> <B><IMG SRC="1029d0790.gif" border=0 width="976" height="121" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <div>  <A NAME="ejercicio_1"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>Ejer</div> <div> <B>Ejercicio 1</B><B>:</B><FONT SIZE="4" COLOR="#FFFFFF" FACE="Georgia" > 1</FONT></div> <div> <IMG SRC="fe6ac9c0.gif" border=0 width="940" height="412" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="ejercicio2"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>Ejercicio2</div> <div> <B>Ejercicio 2</B><B>:</B></div> <div> <B><IMG SRC="10146bb80.gif" border=0 width="875" height="440" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <div>  <A NAME="ejercicio_3"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>Ejercicio 3</div> <div> <B>Ejercicio 3</B><B>:</B></div> <div> <B><IMG SRC="10189cb30.gif" border=0 width="924" height="435" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <div>  <A NAME="ejercicio_4"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>Ejercicio4</div> <div> <B>Ejercicio 4</B><B>:</B></div> <div> <B><IMG SRC="1019d3510.gif" border=0 width="979" height="337" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <div>  <A NAME="ejercicio_5"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>Ejercicio 5</div> <div> <B>Ejercicio 5</B><B>:</B></div> <div> <IMG SRC="ff3a8df0.gif" border=0 width="936" height="479" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="ejercicio_6"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>Ejercicio 6</div> <div> <B>Ejercicio 6</B><B>:</B></div> <div> <IMG SRC="101d075f0.gif" border=0 width="775" height="351" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="ejercicio_7"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>Ejercicio 7</div> <div> <B>Ejercicio 7</B><B>:</B></div> <div> <IMG SRC="15e6085f0.gif" border=0 width="776" height="351" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="ejercicio_8"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>Ejercicio 8</div> <div> <B>Ejercicio 8</B><B>:</B></div> <div> <IMG SRC="1020f32c0.gif" border=0 width="1011" height="812" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> Esta transformada así como la de cos<I>kt</I> también se pueden deducir aplicando directamente la <A HREF="#definición" TITLE="Transformada de Laplace"><U>definición</U></A> y aplicando dos veces integración por partes. Precisamente en el siguiente ejercicio se halla la transformada de Laplace de <I>f </I>(<I>t</I>) = cos<I>kt </I>por el segundo método.</div> <div>  <A NAME="ejercicio_9"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>Ejercicio 9</div> <div> <B>Ejercicio 9</B><B>:</B></div> <div> <IMG SRC="1022d9740.gif" border=0 width="985" height="628" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="apendice"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>Apendice</div> <div> <B>Apéndice:</B></div> <div>  <A NAME="funcion_gamma"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>Funcion gamma</div> <div> <I><B>Definición. La función gamma:</B></I></div> <div> <I><B><IMG SRC="1017ddc50.gif" border=0 width="989" height="453" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></I></div> <div>  <A NAME="continua_por_tramos"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>Continua por tramos</div> <div> <I><B>Definición. Función continua por tramos:</B></I></div> <div> <B><IMG SRC="ffadd340.gif" border=0 width="989" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <div>  <A NAME="orden_exponencial"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A> Orden exponencial</div> <div> <I><B>Definicón. 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