4.1

<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="Georgia" ><B>M</B></FONT><FONT SIZE="4" COLOR="#FFFFFF" FACE="Georgia" ><B>atemáticas de la U</B></FONT><B> (MatU)  | </B><A HREF="http://www.jcbmat.com/" TARGET="_top" TITLE="http://www.jcbmat.com/"><U><B>Home</B></U></A><B> | </B><A HREF="index.htm" TARGET="_top" TITLE="Ecuaciones diferenciales"><U><B>Ecuaciones diferenciales</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="id182.htm" TARGET="_top" TITLE="Z i l l (Ed.7)"><U><B>D. G.. Zill</B></U></A><B> | </B></div> <div> <A HREF="id187.htm" TARGET="_top" TITLE="1.1"><U>1.1</U></A>   |   <A HREF="id282.htm" TARGET="_top" TITLE="Zill 1.2"><U>Zill 1.2</U></A>   |   <A HREF="id283.htm" TARGET="_top" TITLE="1.3"><U>1.3</U></A>   |   <A HREF="id184.htm" TARGET="_top" TITLE="2.2"><U>2.2</U></A>   |   <A HREF="id185.htm" TARGET="_top" TITLE="2.3"><U>2.3</U></A>   |   <A HREF="id186.htm" TARGET="_top" TITLE="2.4"><U>2.4</U></A>   |   <A HREF="id284.htm" TARGET="_top" TITLE="2.5"><U>2.5</U></A>   |   <A HREF="id272.htm" TARGET="_top" TITLE="3.1"><U>3.1</U></A>   |   <B>4.1</B>   |   <A HREF="id285.htm" TARGET="_top" TITLE="4.2"><U>4.2</U></A>   |   <A HREF="id286.htm" TARGET="_top" TITLE="4.3"><U>4.3</U></A>   |   <A HREF="id287.htm" TARGET="_top" TITLE="4.4"><U>4.4</U></A>   |   <A HREF="id288.htm" TARGET="_top" TITLE="4.5"><U>4.5</U></A>   |   <A HREF="id289.htm" TARGET="_top" TITLE="4.6"><U>4.6</U></A>   |   <A HREF="id271.htm" TARGET="_top" TITLE="7.1"><U>7.1</U></A></div> <div> </div> </td> </tr><tr> <td valign="top" rowspan="2" width=160 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> <bR> <bR> </td> <td valign="top" height=432 BGCOLOR="#FFFFFF" TEXT="#080000"> <div> <B>Ejercicios </B><B>4.1</B></div> <div> <B>Teoría preliminar: Ecuaciones lineales</B></div> <bR> <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> <div>  <A NAME="top"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>top</div> <div> <TABLE BORDER="0" CELLPADDING="4" CELLSPACING="1" WIDTH="1003" BORDERCOLORLIGHT="#006666" BORDERCOLORDARK="#006666" FRAME="VOID" RULES="NONE" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP COLSPAN=2 BGCOLOR=#006699 HEIGHT= 23 WIDTH="985"><div> <B>Apuntes: Definiciones, teoremas.</B></div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 23 ><NOBR><div> <A HREF="#pvi" TITLE="4.1"><U><IMG BORDER="0" SRC="Symb075c00b7011866660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Problemas con valores iniciales (PVI)</U></A></div> </NOBR></tD> <tD VALIGN=TOP WIDTH="492"><div> <A HREF="#teorema_de_existencia_y_unicidad" TITLE="4.1"><U><IMG BORDER="0" SRC="Symb075c00b7011811111100.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Teorema de existencia y unicidad</U></A></div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 23 WIDTH="492"><div> <A HREF="#problema_con_valores_en_la_frontera" TITLE="4.1"><U><IMG BORDER="0" SRC="Symb075c00b7011811111100.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Problema con valores en la frontera</U></A></div> </tD> <tD VALIGN=TOP WIDTH="492"><div> <A HREF="#ecuaciones_homogeneas" TITLE="4.1"><U><IMG BORDER="0" SRC="Symb075c00b7011811111100.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Ecuaciones lineales homogéneas</U></A></div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 23 WIDTH="492"><div> <A HREF="#teorema_4_1_2" TITLE="4.1"><U><IMG BORDER="0" SRC="Symb075c00b7011811111100.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Principio de superposición</U></A></div> </tD> <tD VALIGN=TOP WIDTH="492"><div> <A HREF="#definición_4_1_1" TITLE="4.1"><U><IMG BORDER="0" SRC="Symb075c00b7011811111100.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Dependencia e independencia lineal</U></A></div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 23 WIDTH="492"><div> <A HREF="#definición_4_1_2" TITLE="4.1"><U><IMG BORDER="0" SRC="Symb075c00b7011811111100.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Wronskiano</U></A></div> </tD> <tD VALIGN=TOP><NOBR><div> <A HREF="#teorema_4_1_3" TITLE="4.1"><U><IMG BORDER="0" SRC="Symb075c00b7011811111100.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Criterio para soluciones linealmente independientes</U></A></div> </NOBR></tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 23 WIDTH="492"><div> <A HREF="#definición_4_1_3" TITLE="4.1"><U><IMG BORDER="0" SRC="Symb075c00b7011811111100.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Conjunto fundamental de soluciones</U></A></div> </tD> <tD VALIGN=TOP><NOBR><div> <A HREF="#teorema_4_1_4" TITLE="4.1"><U><IMG BORDER="0" SRC="Symb075c00b7011811111100.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Existencia de un conjunto fundamental</U></A></div> </NOBR></tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 23 ><NOBR><div> <A HREF="#teorema_4_1_5" TITLE="4.1"><U><IMG BORDER="0" SRC="Symb075c00b7011811111100.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Solución general de la ED lineal homogénea</U></A></div> </NOBR></tD> <tD VALIGN=TOP><NOBR><div> <A HREF="#teorema_4_1_6" TITLE="4.1"><U><IMG BORDER="0" SRC="Symb075c00b7011811111100.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Solución general de la ED lineal no homogénea</U></A></div> </NOBR></tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 45 WIDTH="492"><div> <A HREF="#función_complementaria" TITLE="4.1"><U><IMG BORDER="0" SRC="Symb075c00b7011811111100.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Función complementaria</U></A></div> </tD> <tD VALIGN=TOP WIDTH="492"><div> <A HREF="#teorema_4_1_7" TITLE="4.1"><U><IMG BORDER="0" SRC="Symb075c00b7011811111100.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Principio de superposición para la ED lineal no homogénea</U></A></div> </tD> </tR> <tR> <tD VALIGN=TOP BGCOLOR=#CCFFFF HEIGHT= 45 WIDTH="492"><div> <A HREF="#enunciados_de_los_ejercicios" TITLE="4.1"><U><IMG BORDER="0" SRC="Symb075c00b7011866660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Ejercicios 4.1</U></A>: Enunciados y soluciones de los problemas</div> </tD> <tD VALIGN=TOP BGCOLOR=#CCFFCC WIDTH="492"><div> <A HREF="#soluciones_de_los_ejercicios" TITLE="4.1"><U><IMG BORDER="0" SRC="Symb075c00b7011811111100.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Soluciones de los ejercicios</U></A></div> </tD> </tR> <tR> <tD VALIGN=TOP COLSPAN=2 BGCOLOR=#FFFFCC HEIGHT= 23 WIDTH="985"><div> <A HREF="#lista_de_reproduccción" TITLE="4.1"><U><IMG BORDER="0" SRC="Symb075c00b7011866660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Lista de reproduccción de los videos en You Tube sobre el Ejercicio 4.1</U></A></div> </tD> </tR> <tR> <tD VALIGN=TOP COLSPAN=2 BGCOLOR=#006699 HEIGHT= 45 WIDTH="985"><div> <U>Nota</U>: Las explicaciones magistrales de esta teoría las puede leer en el libro "Ecuaciones diferenciales con aplicaciones de modelado", Ed. siete o nueve, del Profesor D.G. Zill.</div> </tD> </tR> </TABLE></div> <div>  <A NAME="pvi"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>PVI</div> <div> <B>Problema con valores iniciales PVI para una ED linea</B><FONT SIZE="4" COLOR="#006699" FACE="Bell MT" ><B>l.</B></FONT></div> <div> <FONT SIZE="4" COLOR="#000000" FACE="Georgia" >Para una ecuación diferencial lineal, un </FONT>problema con valores iniciales de <I>n</I>-ésimo orden<FONT SIZE="4" COLOR="#000000" FACE="Georgia" ><B> </B>es </FONT></div> <div> <IMG SRC="146967510.gif" border=0 width="615" height="81" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="teorema_de_existencia_y_unicidad"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>Teorema de existencia y unicidad</div> <div> <IMG SRC="146a6b770.gif" border=0 width="875" height="119" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="problema_con_valores_en_la_frontera"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>Problema con valores en la frontera</div> <div> <B>Problema con valores en la frontera.</B></div> <div> A diferencia de un  <A HREF="#pvi" TITLE="4.1"><U><I>problema con valores iniciales</I></U></A> en donde la variable dependiente y sus derivadas se especifican en un sólo punto, en un <FONT SIZE="4" COLOR="#006699" FACE="Bell MT" >problema con valores en la frontera</FONT> las condiciones se especifican en <I>diferentes puntos</I>. Un problema tal como</div> <div> <IMG SRC="146b8c4f0.gif" border=0 width="396" height="79" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> Se llama un<B> </B><FONT SIZE="4" COLOR="#006699" FACE="Bell MT" ><B>problema con valores en la frontera (PVF)</B></FONT><B> </B>de segundo orden. Los valores prescritos</div> <div> <IMG SRC="146cfc1c0.gif" border=0 width="508" height="28" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> Para una ecuación diferencial de segundo orden , otros pares de condiciones en la frontera podrían ser</div> <div> <IMG SRC="146d80790.gif" border=0 width="384" height="121" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> v    <FONT SIZE="4" COLOR="#111111" FACE="Bell MT" >Un problema con valores en la frontera puede tener varias, una o ninguna solución.</FONT></div> <div>  <A NAME="ecuaciones_homogeneas"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>ecuaciones homogeneas</div> <div> <B>Ecuaciones diferenciales homogéneas y no homogéneas.</B></div> <div> Una ecuación diferencial de <I>n</I>-ésimo orden de la forma</div> <DIV ALIGN="LEFT"></DIV> <div> <IMG SRC="146edb2f0.gif" border=0 width="475" height="47" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> se dice que es <FONT SIZE="4" COLOR="#0066CC" FACE="Bell MT" ><B>homogénea</B></FONT>. Mientras que una ecuación </div> <DIV ALIGN="LEFT"></DIV> <div> <IMG SRC="146ff42f0.gif" border=0 width="500" height="47" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> con <I>g</I>(<I>x</I>) diferente de 0 se dice que es <FONT SIZE="4" COLOR="#0066CC" FACE="Bell MT" ><B>no homogénea</B></FONT>.</div> <div>  <A NAME="teorema_4_1_2"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>Teorema 4.1.2</div> <div> <B>Teorema 4.1.2</B><FONT SIZE="4" COLOR="#0066CC" FACE="Bell MT" ><B>. Principio de superposición; ecuaciones homogéneas.</B></FONT></div> <div> <IMG SRC="147070740.gif" border=0 width="880" height="116" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="definición_4_1_1"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>Definición 4.1.1</div> <div> <B>Definición 4.1.1</B><FONT SIZE="4" COLOR="#0066CC" FACE="Bell MT" ><B>. Dependencia e independencia lineal.</B></FONT></div> <div> <B><IMG SRC="1471248b0.gif" border=0 width="804" height="139" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <div>  <A NAME="definición_4_1_2"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>Definición 4.1.2</div> <div> <B>Definición 4.1.2</B><FONT SIZE="4" COLOR="#0066CC" FACE="Bell MT" ><B>. Wronskiano.</B></FONT></div> <div> <B><IMG SRC="147235c70.gif" border=0 width="821" height="199" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <div>  <A NAME="teorema_4_1_3"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>Teorema 4.1.3</div> <div> <B>Teorema 4.1.3</B><FONT SIZE="4" COLOR="#0066CC" FACE="Bell MT" ><B>. Criterio para soluciones linealmente independientes.</B></FONT></div> <div> <B><IMG SRC="147360570.gif" border=0 width="864" height="87" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <div>  <A NAME="definición_4_1_3"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>Definición 4.1.3</div> <div> <B>Definición 4.1.3</B><FONT SIZE="4" COLOR="#0066CC" FACE="Bell MT" ><B>. Conjunto fundamental de soluciones.</B></FONT></div> <div> <B><IMG SRC="1474444f0.gif" border=0 width="836" height="79" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <div>  <A NAME="teorema_4_1_4"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>Teorema 4.1.4</div> <div> <B>Teorema 4.1.4</B><FONT SIZE="4" COLOR="#0066CC" FACE="Bell MT" ><B>. Existencia de un conjunto fundamental de soluciones.</B></FONT></div> <div> Existe un conjunto fundamental de soluciones para la ecuación diferencial homogénea de <I>n</I>-ésimo orden (<FONT SIZE="4" COLOR="#FF0000" FACE="Bell MT" >6</FONT>) en un intervalo <I>I</I>.</div> <div>  <A NAME="teorema_4_1_5"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>Teorema 4.1.5</div> <div> <FONT SIZE="4" COLOR="#6600FF" FACE="Bell MT" ><B>Teorema 4.1.5</B></FONT><B>. Solución general de la ecuación diferencial homogénea de </B><I><B>n</B></I><B>-ésimo orden.</B></div> <div> <B><IMG SRC="147558900.gif" border=0 width="856" height="144" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <div>  <A NAME="teorema_4_1_6"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>Teorema 4.1.6</div> <div> <FONT SIZE="4" COLOR="#6600FF" FACE="Bell MT" ><B>Teorema 4.1.6</B></FONT><B>. Solución general de la ecuación diferencial no homogénea de </B><I><B>n</B></I><B>-ésimo orden.</B></div> <div> <B><IMG SRC="14765c990.gif" border=0 width="860" height="153" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <div>  <A NAME="función_complementaria"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>Función complementaria</div> <div> v    <FONT SIZE="4" COLOR="#0066CC" FACE="Bell MT" ><B>Función complementaria</B></FONT><FONT SIZE="4" COLOR="#111111" FACE="Bell MT" >. </FONT></div> <div> <IMG SRC="147779c40.gif" border=0 width="889" height="196" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </div> <div>  <A NAME="teorema_4_1_7"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>Teorema 4.1.7</div> <div> <B>Teorema 4.1.7</B><FONT SIZE="4" COLOR="#0066CC" FACE="Bell MT" ><B>. Principio de superposición para las ED lineales no homogéneas.</B></FONT></div> <div> <B><IMG SRC="14786fec0.gif" border=0 width="879" height="236" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <DIV ALIGN="LEFT"></DIV> <div>  <A NAME="enunciados_de_los_ejercicios"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>Enunciados de los ejercicios</div> <div> <B>Enunciados de los ejercicios</B></div> <bR> <div> <FONT SIZE="5" COLOR="#006699" FACE="Bell MT" ><B>S</B></FONT>olución en imagen <IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> y o video <FONT SIZE="3" COLOR="#330033" FACE="Georgia" ><IMG SRC="c4218180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></FONT> de los problemas de los Ejercicios 4.1:</div> <bR> <div> <FONT SIZE="4" COLOR="#CC0033" FACE="Bell MT" ><B> </B></FONT>4.1.1 PROBLEMAS CON VALORES INICIALES Y CON VALORES EN LA FRONTERA</div> <div> <FONT SIZE="4" COLOR="#CC0033" FACE="Bell MT" ><B>    E</B></FONT>n los problemas <FONT SIZE="4" COLOR="#CC0033" FACE="Bell MT" >1</FONT> a <FONT SIZE="4" COLOR="#CC0033" FACE="Bell MT" >4</FONT>, la familia de funciones que se proporciona es la solución general de la ecuación diferencial en el intervalo que se indica. Encuentre un miembro de la familia que sea una solución del problema con valores iniciales.</div> <div> <IMG SRC="1459c91c0.gif" border=0 width="457" height="28" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#soluciones_de_los_ejercicios" TITLE="4.1"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A> <A HREF="https://youtu.be/eqVuZU9d2cU?t=40s" TARGET="_top" TITLE="https://youtu.be/eqVuZU9d2cU?t=40s"><U><IMG SRC="c4218180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <div> <IMG SRC="145d041c0.gif" border=0 width="516" height="28" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#2" TITLE="4.1"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A> <A HREF="https://youtu.be/eqVuZU9d2cU?t=7m19s" TARGET="_top" TITLE="https://youtu.be/eqVuZU9d2cU?t=7m19s"><U><IMG SRC="c4218180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <div> <IMG SRC="1463001c0.gif" border=0 width="512" height="28" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#3" TITLE="4.1"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A> <A HREF="https://youtu.be/eqVuZU9d2cU?t=13m55s" TARGET="_top" TITLE="https://youtu.be/eqVuZU9d2cU?t=13m55s"><U><IMG SRC="c4218180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <div> <IMG SRC="1465811c0.gif" border=0 width="641" height="28" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#4" TITLE="4.1"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A> <A HREF="https://youtu.be/eqVuZU9d2cU?t=21m4s" TARGET="_top" TITLE="https://youtu.be/eqVuZU9d2cU?t=21m4s"><U><IMG SRC="c4218180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <div> <IMG SRC="146867590.gif" border=0 width="871" height="89" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#5" TITLE="4.1"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A> <A HREF="https://youtu.be/u6-NLKMqh_k" TARGET="_top" TITLE="https://youtu.be/u6-NLKMqh_k"><U><IMG SRC="c4218180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <div> <IMG SRC="147e64390.gif" border=0 width="868" height="57" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="#6" TITLE="4.1"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A> <A HREF="https://youtu.be/mi-N4vQt93o" TARGET="_top" TITLE="https://youtu.be/mi-N4vQt93o"><U><IMG SRC="c4218180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <div> <IMG SRC="14805b5f0.gif" border=0 width="859" height="95" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="#7" TITLE="4.1"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A> <A HREF="https://youtu.be/XElR8lCDO1Q" TARGET="_top" TITLE="https://youtu.be/XElR8lCDO1Q"><U><IMG SRC="c4218180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <div> <IMG SRC="148243740.gif" border=0 width="835" height="116" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#8" TITLE="4.1"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A> <A HREF="https://youtu.be/XElR8lCDO1Q" TARGET="_top" TITLE="https://youtu.be/XElR8lCDO1Q"><U><IMG SRC="c4218180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <div> <FONT SIZE="4" COLOR="#CC0033" FACE="Bell MT" ><B>    E</B></FONT>n los problemas <FONT SIZE="4" COLOR="#CC0033" FACE="Bell MT" >9</FONT> y <FONT SIZE="4" COLOR="#CC0033" FACE="Bell MT" >10</FONT>, encuentre un intervalo centrado en <I>x</I> = 0 para el cual el problema con valores iniciales da tiene una solución única.</div> <div> <IMG SRC="14853b1c0.gif" border=0 width="315" height="28" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#9" TITLE="4.1"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A> <FONT SIZE="3" COLOR="#330033" FACE="Georgia" ><A HREF="https://www.youtube.com/watch?v=gUErfcnYdZA" TARGET="_top" TITLE="https://www.youtube.com/watch?v=gUErfcnY"><U><IMG SRC="c4218180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></FONT>       <IMG SRC="1488431c0.gif" border=0 width="323" height="28" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#10" TITLE="4.1"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A> <FONT SIZE="3" COLOR="#330033" FACE="Georgia" ><A HREF="https://www.youtube.com/watch?v=kRK16BkNfLc" TARGET="_top" TITLE="https://www.youtube.com/watch?v=kRK16BkN"><U><IMG SRC="c4218180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></FONT></div> <div> <IMG SRC="148a34940.gif" border=0 width="820" height="148" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="#11" TITLE="4.1"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A> <FONT SIZE="3" COLOR="#330033" FACE="Georgia" ><A HREF="https://www.youtube.com/watch?v=GnGex65hPVA" TARGET="_top" TITLE="https://www.youtube.com/watch?v=GnGex65h"><U><IMG SRC="c4218180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></FONT></div> <div> <IMG SRC="148c052c0.gif" border=0 width="773" height="44" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="#12" TITLE="4.1"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A> <FONT SIZE="3" COLOR="#330033" FACE="Georgia" ><A HREF="https://www.youtube.com/watch?v=w5x9bs_Mp2c" TARGET="_top" TITLE="https://www.youtube.com/watch?v=w5x9bs_M"><U><IMG SRC="c4218180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></FONT></div> <div> <B>    E</B><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" >n los problemas </FONT>13<FONT SIZE="4" COLOR="#000000" FACE="Bell MT" > y </FONT>14<FONT SIZE="4" COLOR="#000000" FACE="Bell MT" > la familia de dos parámetros dada es una solución de de la ecuación diferencial que se indica en el conjuno de todos los reales. Determine si se puede enconrtrar un miembro de la familia que satisfaga las condiciones en la ftontera.</FONT></div> <div> <IMG SRC="148f1d4f0.gif" border=0 width="797" height="79" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="#13" TITLE="4.1"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A><FONT SIZE="3" COLOR="#330033" FACE="Georgia" ><A HREF="https://www.youtube.com/watch?v=nkyrpzQGgYs" TARGET="_top" TITLE="https://www.youtube.com/watch?v=nkyrpzQG"><U><IMG SRC="c4218180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></FONT></div> <div> <IMG SRC="14940b390.gif" border=0 width="779" height="57" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#14" TITLE="4.1"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A><FONT SIZE="3" COLOR="#330033" FACE="Georgia" ><A HREF="https://www.youtube.com/watch?v=DadrMW9vjBM" TARGET="_top" TITLE="https://www.youtube.com/watch?v=DadrMW9v"><U><IMG SRC="c4218180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></FONT></div> <div> 4.1.2 ECUACIONES HOMOGÉNEAS</div> <div> <B>    E</B><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" >n los problemas </FONT>15<FONT SIZE="4" COLOR="#000000" FACE="Bell MT" > a </FONT>22<FONT SIZE="4" COLOR="#000000" FACE="Bell MT" > determine si el conjunto de funciones es linealmente independiente en el intervalo (- infinito, infinito).</FONT></div> <div> <IMG SRC="16774f1d0.gif" border=0 width="335" height="29" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><FONT SIZE="3" COLOR="#330033" FACE="Georgia" ><A HREF="https://www.youtube.com/watch?v=FAsiVyA45j8" TARGET="_top" TITLE="https://www.youtube.com/watch?v=FAsiVyA4"><U><IMG SRC="c4218180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></FONT>          <IMG SRC="1678151c0.gif" border=0 width="277" height="28" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><FONT SIZE="3" COLOR="#330033" FACE="Georgia" ><A HREF="https://www.youtube.com/watch?v=VuLd4NZ8beQ" TARGET="_top" TITLE="https://www.youtube.com/watch?v=VuLd4NZ8"><U><IMG SRC="c4218180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></FONT></div> <div> <IMG SRC="16795b1c0.gif" border=0 width="347" height="28" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><FONT SIZE="3" COLOR="#330033" FACE="Georgia" ><A HREF="https://www.youtube.com/watch?v=zbRh5f_LuzE" TARGET="_top" TITLE="https://www.youtube.com/watch?v=zbRh5f_L"><U><IMG SRC="c4218180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></FONT>          <IMG SRC="167a571c0.gif" border=0 width="343" height="28" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><FONT SIZE="3" COLOR="#330033" FACE="Georgia" ><A HREF="https://www.youtube.com/watch?v=-QA4kIkTMzw" TARGET="_top" TITLE="https://www.youtube.com/watch?v=-QA4kIkT"><U><IMG SRC="c4218180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></FONT></div> <div> <IMG SRC="167b441b0.gif" border=0 width="324" height="27" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><FONT SIZE="3" COLOR="#330033" FACE="Georgia" ><A HREF="https://www.youtube.com/watch?v=L5E6TpUlmMo" TARGET="_top" TITLE="https://www.youtube.com/watch?v=L5E6TpUl"><U><IMG SRC="c4218180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></FONT>         <IMG SRC="167cff1c0.gif" border=0 width="255" height="28" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><FONT SIZE="3" COLOR="#330033" FACE="Georgia" ><A HREF="https://www.youtube.com/watch?v=06zoiFIXQwk" TARGET="_top" TITLE="https://www.youtube.com/watch?v=06zoiFIX"><U><IMG SRC="c4218180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></FONT>         </div> <div> <IMG SRC="167d301c0.gif" border=0 width="304" height="28" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><FONT SIZE="3" COLOR="#330033" FACE="Georgia" ><A HREF="https://www.youtube.com/watch?v=ft8CPtE-Png" TARGET="_top" TITLE="https://www.youtube.com/watch?v=ft8CPtE-"><U><IMG SRC="c4218180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></FONT>         <IMG SRC="167e551c0.gif" border=0 width="341" height="28" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><FONT SIZE="3" COLOR="#330033" FACE="Georgia" ><A HREF="https://www.youtube.com/watch?v=BNTbSLNAHng" TARGET="_top" TITLE="https://www.youtube.com/watch?v=BNTbSLNA"><U><IMG SRC="c4218180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></FONT></div> <div> <FONT SIZE="4" COLOR="#CC0033" FACE="Bell MT" ><B>    E</B></FONT>n los problemas <FONT SIZE="4" COLOR="#CC0033" FACE="Georgia" >23</FONT> a <FONT SIZE="4" COLOR="#CC0033" FACE="Georgia" >30</FONT> compruebe que las funciones dadas  forman un conjunto fundamental de soluciones de la ecuación diferencial en el intervalo que se indica. Forme la solución general.</div> <div> <IMG SRC="167f571d0.gif" border=0 width="343" height="29" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><FONT SIZE="3" COLOR="#330033" FACE="Georgia" ><A HREF="https://www.youtube.com/watch?v=zvl2Xf8tEpo" TARGET="_top" TITLE="https://www.youtube.com/watch?v=zvl2Xf8t"><U><IMG SRC="c4218180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></FONT>          <IMG SRC="1680651c0.gif" border=0 width="357" height="28" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><FONT SIZE="3" COLOR="#330033" FACE="Georgia" ><A HREF="https://www.youtube.com/watch?v=razUuQPfS_8" TARGET="_top" TITLE="https://www.youtube.com/watch?v=razUuQPf"><U><IMG SRC="c4218180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></FONT></div> <div> <FONT SIZE="4" COLOR="#330033" FACE="Georgia" ><IMG SRC="1681a51c0.gif" border=0 width="421" height="28" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></FONT><A HREF="https://www.youtube.com/watch?v=BQhhT9qL_lg" TARGET="_top" TITLE="https://www.youtube.com/watch?v=BQhhT9qL"><U><IMG SRC="c4218180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>         <IMG SRC="16826b1f0.gif" border=0 width="363" height="31" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="https://www.youtube.com/watch?v=qX_KUaD2za8" TARGET="_top" TITLE="https://www.youtube.com/watch?v=qX_KUaD2"><U><IMG SRC="c4218180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <div> <IMG SRC="16834f1c0.gif" border=0 width="335" height="28" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="https://www.youtube.com/watch?v=P_szwKGZ84E" TARGET="_top" TITLE="https://www.youtube.com/watch?v=P_szwKGZ"><U><IMG SRC="c4218180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>         <IMG SRC="1684a51c0.gif" border=0 width="421" height="28" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="https://www.youtube.com/watch?v=K12fa1Nr0zA" TARGET="_top" TITLE="https://www.youtube.com/watch?v=K12fa1Nr"><U><IMG SRC="c4218180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <div> <IMG SRC="1685c91c0.gif" border=0 width="457" height="28" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="https://www.youtube.com/watch?v=oJXl6eTFZP4" TARGET="_top" TITLE="https://www.youtube.com/watch?v=oJXl6eTF"><U><IMG SRC="c4218180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>         <IMG SRC="16866d1c0.gif" border=0 width="365" height="28" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="https://www.youtube.com/watch?v=VQkqItp4gMM" TARGET="_top" TITLE="https://www.youtube.com/watch?v=VQkqItp4"><U><IMG SRC="c4218180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <div> 4.1.3 ECUACIONES NO HOMOGÉNEAS</div> <div> <FONT SIZE="4" COLOR="#CC0033" FACE="Bell MT" ><B>    E</B></FONT>n los problemas <FONT SIZE="4" COLOR="#CC0033" FACE="Georgia" >31</FONT> a <FONT SIZE="4" COLOR="#CC0033" FACE="Georgia" >34</FONT> compruebe que dada la familia de soluciones de dos parámetros, se trata de la solución general de la ecuación diferencial no homogénea en el intervalo indicado.</div> <div> <IMG SRC="1687931c0.gif" border=0 width="403" height="28" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><FONT SIZE="3" COLOR="#330033" FACE="Georgia" ><A HREF="https://www.youtube.com/watch?v=LzzJOLCdlgM" TARGET="_top" TITLE="https://www.youtube.com/watch?v=LzzJOLCd"><U><IMG SRC="c4218180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></FONT></div> <div> <IMG SRC="1688191c0.gif" border=0 width="537" height="28" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><FONT SIZE="3" COLOR="#330033" FACE="Georgia" ><A HREF="https://www.youtube.com/watch?v=hBY0ranXxr8" TARGET="_top" TITLE="https://www.youtube.com/watch?v=hBY0ranX"><U><IMG SRC="c4218180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></FONT></div> <div> <IMG SRC="16892b1c0.gif" border=0 width="555" height="28" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><FONT SIZE="3" COLOR="#330033" FACE="Georgia" ><A HREF="https://www.youtube.com/watch?v=wIsU3JgB7VU" TARGET="_top" TITLE="https://www.youtube.com/watch?v=wIsU3JgB"><U><IMG SRC="c4218180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></FONT></div> <div> <IMG SRC="168af12d0.gif" border=0 width="497" height="45" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><FONT SIZE="3" COLOR="#330033" FACE="Georgia" ><A HREF="https://www.youtube.com/watch?v=nhRLzUhlPwk" TARGET="_top" TITLE="https://www.youtube.com/watch?v=nhRLzUhl"><U><IMG SRC="c4218180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></FONT></div> <div> <IMG SRC="168b607c0.gif" border=0 width="864" height="124" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="https://www.youtube.com/watch?v=3ZGnqPl1BgE" TARGET="_top" TITLE="https://www.youtube.com/watch?v=3ZGnqPl1"><U><IMG SRC="c4218180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <div> <IMG SRC="168c0ccc0.gif" border=0 width="524" height="204" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="https://www.youtube.com/watch?v=vTDemCe4wfI" TARGET="_top" TITLE="https://www.youtube.com/watch?v=vTDemCe4"><U><IMG SRC="c4218180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <DIV ALIGN="LEFT"></DIV> <div>  <A NAME="lista_de_reproduccción"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>Lista de reproduccción</div> <div> <B>Lista de reproduccción de los videos en You Tube sobre el Ejercicio 4.1</B></div> <bR> <div> <B> <iframe width="560" height="315" src="https://www.youtube.com/embed/videoseries?list=PLq_SQwoOZHo1-GPxpKQZxL8kvFKlfy7Rp" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>  </B></div> <div>  <A NAME="soluciones_de_los_ejercicios"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><FONT SIZE="1" COLOR="#FFFFFF" FACE="Bell MT" >Soluciones de los ejercicios</FONT></div> <div> <B>Soluciones de los ejercicios</B></div> <div>  <A NAME="1"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>1</div> <div> <B><IMG SRC="145c70510.gif" border=0 width="880" height="337" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <div>  <A NAME="2"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>2</div> <div> <B><IMG SRC="145e70530.gif" border=0 width="880" height="339" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <div>  <A NAME="3"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>3</div> <div> <B><IMG SRC="146275440.gif" border=0 width="885" height="324" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <div>  <A NAME="4"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>4</div> <div> <B><IMG SRC="14645c930.gif" border=0 width="860" height="403" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <div>  <A NAME="5"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>5</div> <div> <B><IMG SRC="146770a70.gif" border=0 width="880" height="423" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <div>  <A NAME="6"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>6</div> <div> <B><IMG SRC="147d738f0.gif" border=0 width="883" height="399" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <div>  <A NAME="7"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>7</div> <div> <B><IMG SRC="147f84d40.gif" border=0 width="900" height="468" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <div>  <A NAME="8"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>8</div> <div> <B><IMG SRC="148188440.gif" border=0 width="904" height="1092" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <div>  <A NAME="9"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>9</div> <div> <B><IMG SRC="14842daf0.gif" border=0 width="813" height="431" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <div>  <A NAME="10"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>10</div> <div> <B><IMG SRC="14873d770.gif" border=0 width="829" height="375" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <div>  <A NAME="11"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>11</div> <div> <B><IMG SRC="148964b70.gif" border=0 width="868" height="1207" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <div>  <A NAME="12"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>12</div> <div> <B><IMG SRC="148b47b30.gif" border=0 width="839" height="435" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <div>  <A NAME="13"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>13</div> <div> <B><IMG SRC="148e20150.gif" border=0 width="800" height="533" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <div> <B><IMG SRC="14904cff0.gif" border=0 width="844" height="255" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <div> <B><IMG SRC="149134680.gif" border=0 width="820" height="360" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <div> <B><IMG SRC="1492171c0.gif" border=0 width="791" height="284" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <div>  <A NAME="14"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>14</div> <div> <B><IMG SRC="1493536f0.gif" border=0 width="851" height="367" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <div> <B><IMG SRC="149519d70.gif" border=0 width="793" height="215" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <div> <IMG SRC="149660750.gif" border=0 width="864" height="373" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <IMG SRC="149705670.gif" border=0 width="773" height="359" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <DIV ALIGN="LEFT"></DIV> <div>  <script async src="https://pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script>  <ins class="adsbygoogle" style="display:inline-block;width:970px;height:250px" data-ad-client="ca-pub-1578463498418701" data-ad-slot="8553279920"></ins> <script> (adsbygoogle = window.adsbygoogle || []).push({}); </script>  </div> </td> <td valign="top" rowspan="2" width=105 BGCOLOR="#FFFFFF" TEXT="#080000"> <div> <A HREF="https://www.youtube.com/user/MatematicaAplicadas" TARGET="_top" TITLE="https://www.youtube.com/user/MatematicaA"><U><IMG SRC="154c64580.gif" border=0 width="100" height="600" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <div> <A HREF="id279_m.htm#top" TARGET="TRLX_Middle" TITLE="4.1"><U><B>Arriba</B></U></A></div> </td> </tr><tr> <td valign="top" height=24 BGCOLOR="#003366" TEXT="#F7F7FF"> <bR> </td> </tr></table></body>