Zill 1.2

<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>   |   <B>Zill 1.2</B>   |   <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>   |   <A HREF="id279.htm" TARGET="_top" TITLE="4.1"><U>4.1</U></A>   |   <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> </td> <td valign="top" height=432 BGCOLOR="#FFFFFF" TEXT="#080000"> <div> <B>Ejercicios </B><B>1.2</B></div> <div> <B>Problemas con valores iniciales</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> <bR> <div> <TABLE BORDER="0" CELLPADDING="4" CELLSPACING="1" WIDTH="903" 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="885"><div>  <A NAME="apuntes"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><FONT SIZE="4" COLOR="#FFFFFF" FACE="Georgia" ><B>Apuntes: Definiciones, teoremas.</B></FONT></div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 23 ><NOBR><div> <A HREF="#pvi" TITLE="Zill 1.2"><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><NOBR><div> <A HREF="#teorema_de_existencia_y_unicidad" TITLE="Zill 1.2"><U><IMG BORDER="0" SRC="Symb075c00b7011866660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Teorema 1.2.1</U></A>: Existencia de una solución única</div> </NOBR></tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 23 WIDTH="442"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="23" ALIGN="bottom" WIDTH="442" HSPACE="0" VSPACE="0"></tD> <tD VALIGN=TOP WIDTH="442"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="23" ALIGN="bottom" WIDTH="442" HSPACE="0" VSPACE="0"></tD> </tR> <tR> <tD VALIGN=TOP BGCOLOR=#CCFFFF HEIGHT= 45 WIDTH="442"><div> <A HREF="#ejercicios_1_2_1" TITLE="Zill 1.2"><U><IMG BORDER="0" SRC="Symb075c00b7011866660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Ejercicios 1.2</U></A>: Enunciados y soluciones de los problemas</div> </tD> <tD VALIGN=TOP BGCOLOR=#CCFFCC WIDTH="442"><div> <A HREF="#soluciones" TITLE="Zill 1.2"><U><IMG BORDER="0" SRC="Symb075c00b7011866660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Ejercicios 1.2</U></A>: Lista de reproduccción de los videos en You Tube</div> </tD> </tR> <tR> <tD VALIGN=TOP COLSPAN=2 BGCOLOR=#006699 HEIGHT= 45 WIDTH="885"><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><IMG BORDER="0" SRC="Symb075c00b7011899330000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Problemas</B><B> con valores iniciales de</B><B> </B><I><B>n-</B></I><B>ésimo</B><B> orden (PVI).</B></div> <div> <IMG SRC="15b1a3cf0.gif" border=0 width="931" height="207" 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> <B><IMG BORDER="0" SRC="Symb075c00b7011899330000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Teorema 1.2.1: Existencia de una solución única.</B></div> <DIV ALIGN="LEFT"></DIV> <div> <B><IMG SRC="1806daed0.gif" border=0 width="218" height="237" ALT="Imagen de Paintbrush" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <div> <B><IMG SRC="15d9b1c40.gif" border=0 width="945" height="196" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <DIV ALIGN="LEFT"></DIV> <div>  <A NAME="ejercicios_1_2_1"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><FONT SIZE="4" COLOR="#FFFFFF" FACE="Bell MT" ><B>Ejercicios 1.2</B></FONT></div> <div> <B>Enunciados y enlaces a las soluciones de los problemas de los Ejercicios 1.2</B></div> <bR> <div> <FONT SIZE="5" COLOR="#006600" 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 1.2:</div> <bR> <div> <B><IMG SRC="15b2ff570.gif" border=0 width="1023" height="87" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <div> <FONT SIZE="4" COLOR="#009900" FACE="Bell MT" ><B><IMG SRC="15b36d300.gif" border=0 width="109" height="48" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </B></FONT><A HREF="https://youtu.be/o5nZvVaGo7g" TARGET="_top" TITLE="https://youtu.be/o5nZvVaGo7g"><U><IMG SRC="c4218180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>        <IMG SRC="15b46b1c0.gif" border=0 width="107" height="28" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="https://youtu.be/o5nZvVaGo7g" TARGET="_top" TITLE="https://youtu.be/o5nZvVaGo7g"><U><IMG SRC="c4218180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <div> <IMG SRC="1632ad570.gif" border=0 width="941" height="87" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <FONT SIZE="3" COLOR="#330033" FACE="Georgia" ><IMG SRC="15b661300.gif" border=0 width="97" height="48" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </FONT><A HREF="#3" TITLE="Zill 1.2"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>        <IMG SRC="15bd6f300.gif" border=0 width="111" height="48" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#4" TITLE="Zill 1.2"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="15bf5c1c0.gif" border=0 width="92" height="28" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#5" TITLE="Zill 1.2"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="15c173340.gif" border=0 width="115" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#6" TITLE="Zill 1.2"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <div> <IMG SRC="15c2a7530.gif" border=0 width="935" height="83" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <IMG SRC="15c4bc1c0.gif" border=0 width="188" height="28" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#7" TITLE="Zill 1.2"><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" ><IMG SRC="c4218180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></FONT>       <IMG SRC="15c6e31c0.gif" border=0 width="227" height="28" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#8" TITLE="Zill 1.2"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="15c8cc340.gif" border=0 width="204" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#9" TITLE="Zill 1.2"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       </div> <div> <FONT SIZE="4" COLOR="#000000" FACE="Bell MT" ><IMG SRC="15ddf1340.gif" border=0 width="241" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </FONT><A HREF="https://youtu.be/PrMvx7scFzo" TARGET="_top" TITLE="https://youtu.be/PrMvx7scFzo"><U><IMG SRC="c4218180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>      </div> <div> <IMG SRC="15e0d44f0.gif" border=0 width="980" height="79" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <IMG SRC="15deb51c0.gif" border=0 width="181" height="28" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="https://youtu.be/Ef8LVUr4Nfo" TARGET="_top" TITLE="https://youtu.be/Ef8LVUr4Nfo"><U><IMG SRC="c4218180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="15dfb11c0.gif" border=0 width="177" height="28" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="https://youtu.be/I7SJi14Xvhw" TARGET="_top" TITLE="https://youtu.be/I7SJi14Xvhw"><U><IMG SRC="c4218180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="15e1d81c0.gif" border=0 width="216" height="28" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="https://youtu.be/Tfi6Syl-X6A" TARGET="_top" TITLE="https://youtu.be/Tfi6Syl-X6A"><U><IMG SRC="c4218180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="15e2b81c0.gif" border=0 width="184" height="28" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="https://youtu.be/2d9lcFcmZW4" TARGET="_top" TITLE="https://youtu.be/2d9lcFcmZW4"><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> <B> <A NAME="soluciones"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A></B><B>Soluciones</B></div> <div> <B>Soluciones </B><B>de los problemas de los Ejercicios 1.2</B></div> <bR> <div> <B> <iframe width="853" height="480" src="https://www.youtube.com/embed/videoseries?list=PLq_SQwoOZHo2kMlbbQmM6l2dUX9lVMxjc" frameborder="0" allowfullscreen></iframe>  </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> <IMG SRC="15b7dd840.gif" border=0 width="989" height="388" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></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> <IMG SRC="15bccd8d0.gif" border=0 width="973" height="397" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></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> <IMG SRC="15be915f0.gif" border=0 width="913" height="351" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></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> <IMG SRC="15c0f5a10.gif" border=0 width="1013" height="417" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></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> <IMG SRC="15c327640.gif" border=0 width="807" height="356" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></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> <IMG SRC="15c527640.gif" border=0 width="807" height="356" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></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> <IMG SRC="15c727890.gif" border=0 width="807" height="649" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <bR> <bR> <DIV ALIGN="LEFT"></DIV> <div> <IMG SRC="15528c8c0.gif" border=0 width="140" height="140" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div> <A HREF="http://twitter.com/Calculo21" TARGET="_top" TITLE="http://twitter.com/Calculo21"><U><IMG SRC="97d28280.jpg" border=0 width="40" height="40" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A><A HREF="http://www.facebook.com/juan.beltran.58511" TARGET="_top" TITLE="http://www.facebook.com/juan.beltran.585"><U><IMG SRC="97e28280.gif" border=0 width="40" height="40" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A><A HREF="https://www.youtube.com/user/MatematicaAplicadas" TARGET="_top" TITLE="https://www.youtube.com/user/MatematicaA"><U><IMG SRC="97f28280.jpg" border=0 width="40" height="40" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <bR> <div> Por: <FONT SIZE="4" COLOR="#FF9900" FACE="Georgia" >Juan Carlos </FONT><FONT SIZE="4" COLOR="#0000FF" FACE="Georgia" >Beltrán</FONT> <FONT SIZE="4" COLOR="#FF0000" FACE="Georgia" >Beltrán </FONT></div> <bR> <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> <bR> <div> <A HREF="http://jcbmat.com/" TARGET="_top" TITLE="http://jcbmat.com/"><U><B><IMG SRC="1558e5530.gif" border=0 width="741" height="83" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A></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="id282_m.htm#ejercicios_1_2_1" TARGET="TRLX_Middle" TITLE="Zill 1.2"><U><B>Enunciados</B></U></A></div> </td> </tr><tr> <td valign="top" height=24 BGCOLOR="#003366" TEXT="#F7F7FF"> <div> <B> <form action="http://www.google.com.co" id="cse-search-box" target="_blank"> <div> <input type="hidden" name="cx" value="partner-pub-6266400581769473:6497222116" /> <input type="hidden" name="ie" value="UTF-8" /> <input type="text" name="q" size="49" /> <input type="submit" name="sa" value="Buscar en MatU" /> </div> </form> <script type="text/javascript" src="http://www.google.com.co/coop/cse/brand?form=cse-search-box&lang=es"></script>  </B></div> </td> </tr></table></body>