3.1

<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> <B>| </B><A HREF="index.htm" TARGET="_top" TITLE="Ecuaciones diferenciales"><U><B>Ecuaciones diferenciales</B></U></A><B>  |  </B><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="id31.htm" TARGET="_top" TITLE="B o y c e y D i P r i m a"><U><B>Boyce y DiPrima</B></U></A><B>  |</B></div> <div> <A HREF="id131.htm" TARGET="_top" TITLE="1.1"><U>1.1</U></A>   |   <A HREF="id132.htm" TARGET="_top" TITLE="2.1"><U>2.1</U></A>   |   <A HREF="id133.htm" TARGET="_top" TITLE="2.2"><U>2.2</U></A>   |   <A HREF="id135.htm" TARGET="_top" TITLE="2.3"><U>2.3</U></A>   |   <A HREF="id137.htm" TARGET="_top" TITLE="2.4"><U>2.4</U></A>   |   <A HREF="id136.htm" TARGET="_top" TITLE="2.5"><U>2.5</U></A>   |   <A HREF="id274.htm" TARGET="_top" TITLE="2.10"><U>2.10</U></A>   |   <B>3.1</B>   |   <A HREF="id269.htm" TARGET="_top" TITLE="3.2"><U>3.2</U></A>   |   <A HREF="id140.htm" TARGET="_top" TITLE="7.2(a)"><U>7.2(a)</U></A>   |   <A HREF="id143.htm" TARGET="_top" TITLE="8.1"><U>8.1</U></A></div> <div> </div> </td> </tr><tr> <td valign="top" 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=456 BGCOLOR="#FFFFFF" TEXT="#080000"> <div> <B>Capítulo 3: Ecuaciones </B><B>lin</B><B>eales de segundo oreden</B></div> <div> <B>Sección 3.1: "Ecuaciones homogéneas con coeficientes constantes"</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> <B>Apuntes</B></div> <div>  <A NAME="ecuación_diferencial_de_segundo_orden"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>Ecuación diferencial de segundo orden</div> <div> <FONT SIZE="4" COLOR="#006633" FACE="Georgia" ><B><IMG BORDER="0" SRC="Symb075c00b7011833660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Ecuación diferencial de segundo orden.</B></FONT> Una ecuación diferencial ordinaria de segun orden tiene la forma</div> <DIV ALIGN="LEFT"></DIV> <div> <IMG SRC="14e6cf340.gif" border=0 width="207" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="ed_lineal_de_segundo_orden"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>ed lineal de segundo orden</div> <div> <B><IMG BORDER="0" SRC="Symb075c00b7011833660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Ecuación diferencial lineal de segundo orden.</B><FONT SIZE="3" COLOR="#006633" FACE="Georgia" ><B> </B></FONT><FONT SIZE="4" COLOR="#000000" FACE="Georgia" >La forma general de una ecuación diferencial lineal de segundo orden es</FONT></div> <DIV ALIGN="LEFT"></DIV> <div> <IMG SRC="14ebd8300.gif" border=0 width="728" height="48" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> Si se divide cada término en (<FONT SIZE="4" COLOR="#FF0000" FACE="Georgia" >2</FONT>) entre <I>P</I>(<I>x</I>), se obtiene</div> <bR> <DIV ALIGN="LEFT"></DIV> <div> <IMG SRC="14e80f300.gif" border=0 width="527" height="48" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="problema_con_valor_inicial"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>Problema con valor inicial</div> <div> <FONT SIZE="4" COLOR="#006633" FACE="Georgia" ><B><IMG BORDER="0" SRC="Symb075c00b7011833660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Problema con valor inicial.</B></FONT> Un problema con valor inicial consta de una ecuación diferencial como (<FONT SIZE="4" COLOR="#FF0000" FACE="Georgia" >1</FONT>), (<FONT SIZE="4" COLOR="#FF0000" FACE="Georgia" >2</FONT>) o (<FONT SIZE="4" COLOR="#FF0000" FACE="Georgia" >3</FONT>) junto con un par de condiciones iniciales de la forma </div> <DIV ALIGN="LEFT"></DIV> <div> <IMG SRC="14e9fc1c0.gif" border=0 width="252" height="28" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="ecuación_homogenea"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>ecuación homogenea</div> <div> <FONT SIZE="4" COLOR="#006633" FACE="Georgia" ><B><IMG BORDER="0" SRC="Symb075c00b7011833660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Ecuación diferencial lineal homogénea de segundo orden.</B></FONT> Se dice que una ecuación lineal de segundo orden es <I><B>homogénea </B></I>si el término G(<I>x</I>) de la ecuación (<FONT SIZE="4" COLOR="#FF0000" FACE="Georgia" >2</FONT>), o el término g(<I>x</I>) de la (<FONT SIZE="4" COLOR="#FF0000" FACE="Georgia" >3</FONT>), es cero para toda <I>x</I>. Esto es, si</div> <DIV ALIGN="LEFT"></DIV> <div> <IMG SRC="14ea10190.gif" border=0 width="272" height="25" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> Si los coeficientes de la ecuación (<FONT SIZE="4" COLOR="#FF0000" FACE="Georgia" >5</FONT>) son constantes, se tiene la ED lineal homogénea de segundo orden con coeficientes contantes,</div> <DIV ALIGN="LEFT"></DIV> <div> <IMG SRC="14ecbf190.gif" border=0 width="191" height="25" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="ecuación_característica"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>Ecuación característica</div> <div> <B><IMG BORDER="0" SRC="Symb075c00b7011833660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Ecuación característica. </B></div> <div> <IMG SRC="14ed1c6f0.gif" border=0 width="796" height="111" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <DIV ALIGN="LEFT"></DIV> <div> <IMG SRC="14eeab1b0.gif" border=0 width="171" height="27" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> La ecuación cuadrática dada en (<FONT SIZE="4" COLOR="#FF0000" FACE="Georgia" >7</FONT>) se conoce como la <I><B>ecuación característica</B></I> asociada a la ED (<FONT SIZE="4" COLOR="#FF0000" FACE="Georgia" >6</FONT>). Por supuesto, la ecuación característica tiene dos raíces que pueden ser reales y distintas, reales y repetidas o complejas conjugadas.</div> <div>  <A NAME="solución_general"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>Solución general</div> <div> <B><IMG BORDER="0" SRC="Symb075c00b7011833660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Solución general de la ecuación deiferencial lineal homogénea de segundo orden con coeficientes constantes cuando la ecuación característica tiene dos raíces reales distintas.</B><FONT SIZE="4" COLOR="#000000" FACE="Georgia" > </FONT></div> <div> Si las raíces de la ecuación característica (<FONT SIZE="4" COLOR="#FF0000" FACE="Georgia" >7</FONT>) son reales y distintas, la solución general de la ecuación diferencial (<FONT SIZE="4" COLOR="#FF0000" FACE="Georgia" >6</FONT>) es de la forma </div> <DIV ALIGN="LEFT"></DIV> <div> <IMG SRC="14ef881d0.gif" border=0 width="136" height="29" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> donde <I>r</I><FONT SIZE="3" COLOR="#000000" FACE="Georgia" >1</FONT> y <I>r</I><FONT SIZE="3" COLOR="#000000" FACE="Georgia" >2</FONT> son las raíces de la ecuación característica.</div> <bR> <bR> <div>  <A NAME="enunciados"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>enunciados</div> <DIV ALIGN="LEFT"></DIV> <div> <B>E n u n c i a d o s   d e   l o s   p r o b l e m a s</B></div> <bR> <div> <FONT SIZE="4" COLOR="#0099FF" FACE="Bell MT" ><B>E</B></FONT><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" >n cada uno de los problemas</FONT> <FONT SIZE="4" COLOR="#CC0033" FACE="Bell MT" >1</FONT> <FONT SIZE="4" COLOR="#111111" FACE="Bell MT" >a</FONT> <FONT SIZE="4" COLOR="#CC0033" FACE="Bell MT" >8</FONT><FONT SIZE="4" COLOR="#111111" FACE="Bell MT" >, halle la solución general de la ecuación diferencial dada.</FONT></div> <div> <IMG SRC="14d58b150.gif" border=0 width="139" height="21" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#1" TITLE="3.1"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>        <IMG SRC="14d680150.gif" border=0 width="128" height="21" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#2" TITLE="3.1"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="14d785150.gif" border=0 width="133" height="21" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#3" TITLE="3.1"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="14d88d150.gif" border=0 width="141" height="21" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#4" TITLE="3.1"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>        </div> <div> <IMG SRC="14d96c150.gif" border=0 width="108" height="21" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#5" TITLE="3.1"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="14da69150.gif" border=0 width="105" height="21" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#6" TITLE="3.1"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="14db8c150.gif" border=0 width="140" height="21" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#7" TITLE="3.1"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="14dc8c150.gif" border=0 width="140" height="21" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#8" TITLE="3.1"><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="#0099FF" FACE="Bell MT" ><B>E</B></FONT>n cada uno de los problemas <FONT SIZE="4" COLOR="#CC0033" FACE="Bell MT" >9</FONT> a <FONT SIZE="4" COLOR="#CC0033" FACE="Bell MT" >14</FONT>, encuentre la solución del problema con valor inicial dado. Trace la gráfica de la solución y describa su comportamiento al crecer <I>x</I>.</div> <div> <IMG SRC="14dd18170.gif" border=0 width="280" height="23" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#9" TITLE="3.1"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>          <IMG SRC="14de39170.gif" border=0 width="313" height="23" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#10" TITLE="3.1"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <div> <IMG SRC="14df30170.gif" border=0 width="304" height="23" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#11" TITLE="3.1"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A> <A HREF="https://youtu.be/iY5sV0aqjKk" TARGET="_top" TITLE="https://youtu.be/iY5sV0aqjKk"><U><IMG SRC="c4218180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>           <IMG SRC="14e018170.gif" border=0 width="280" height="23" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#12" TITLE="3.1"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <div> <IMG SRC="14e125170.gif" border=0 width="293" height="23" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#13" TITLE="3.1"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>           <IMG SRC="14e211150.gif" border=0 width="273" height="21" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#14" TITLE="3.1"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <div> <IMG SRC="14e4c8310.gif" border=0 width="712" height="49" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <FONT SIZE="4" COLOR="#111111" FACE="Bell MT" ><A HREF="#15" TITLE="3.1"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></FONT></div> <div> <IMG SRC="14e584310.gif" border=0 width="644" height="49" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <FONT SIZE="4" COLOR="#111111" FACE="Bell MT" ><A HREF="#16" TITLE="3.1"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></FONT></div> <div> <IMG SRC="1392b46d0.gif" border=0 width="948" height="109" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <FONT SIZE="4" COLOR="#0099FF" FACE="Bell MT" ><B>E</B></FONT>n cada uno de los problemas <FONT SIZE="4" COLOR="#CC0033" FACE="Bell MT" >17</FONT> a <FONT SIZE="4" COLOR="#CC0033" FACE="Bell MT" >22</FONT>, aplique esta sustitución para resolver la ecuación dada.</div> <div> <IMG SRC="1394d4190.gif" border=0 width="212" height="25" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#17" TITLE="3.1"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A> <A HREF="https://youtu.be/p1j8u_-EbKs" TARGET="_top" TITLE="https://youtu.be/p1j8u_-EbKs"><U><IMG SRC="c4218180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="1396a4150.gif" border=0 width="164" height="21" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#18" TITLE="3.1"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="14f095200.gif" border=0 width="149" height="32" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#19" TITLE="3.1"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <div> <IMG SRC="14f3c1200.gif" border=0 width="193" height="32" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="#20" TITLE="3.1"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A> <A HREF="https://youtu.be/rQOiskh4kME" TARGET="_top" TITLE="https://youtu.be/rQOiskh4kME"><U><IMG SRC="c4218180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="14f6871c0.gif" border=0 width="135" height="28" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="https://youtu.be/tVrih3ZK4yE" TARGET="_top" TITLE="https://youtu.be/tVrih3ZK4yE"><U><IMG SRC="c4218180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="14f883200.gif" border=0 width="131" height="32" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="https://youtu.be/f8ISn6oeR5g" TARGET="_top" TITLE="https://youtu.be/f8ISn6oeR5g"><U><IMG SRC="c4218180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <div>   <A NAME="soluciones_1"><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</FONT></div> <DIV ALIGN="LEFT"></DIV> <div> <B>Soluciones de los problemas</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> <IMG SRC="135951540.gif" border=0 width="593" height="340" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></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> <IMG SRC="1366e83d0.gif" border=0 width="488" height="317" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></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="136813730.gif" border=0 width="531" height="371" 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="136a2b590.gif" border=0 width="555" height="345" 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="136c13570.gif" border=0 width="531" height="343" 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="136ee85b0.gif" border=0 width="488" height="347" 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> <B><IMG SRC="1370457c0.gif" border=0 width="837" height="380" 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> <IMG SRC="1372405d0.gif" border=0 width="832" height="349" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div> <FONT SIZE="4" COLOR="#0099FF" FACE="Bell MT" ><B>E</B></FONT>n cada uno de los problemas <FONT SIZE="4" COLOR="#CC0033" FACE="Bell MT" >9</FONT> a <FONT SIZE="4" COLOR="#CC0033" FACE="Bell MT" >14</FONT>, encuentre la solución del problema con valor inicial dado. Trace la gráfica de la solución y describa su comportamiento al crecer <I>x</I>.</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> <TABLE BORDER="0" CELLPADDING="2" CELLSPACING="1" WIDTH="803" BORDERCOLORLIGHT="#C0C0C0" BORDERCOLORDARK="#808080" FRAME="BOX" RULES="ALL" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP HEIGHT= 401 ><NOBR><div> <IMG SRC="1374af890.gif" border=0 width="431" height="393" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> <tD VALIGN=TOP><NOBR><bR> <bR> <bR> <div> <IMG SRC="6d0350c0.gif" border=0 width="309" height="268" ALT="Imagen de mapa de bits" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> </tR> </TABLE></div> <div> <IMG SRC="1375589c0.gif" border=0 width="856" height="412" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></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> <TABLE BORDER="0" CELLPADDING="2" CELLSPACING="1" WIDTH="803" BORDERCOLORLIGHT="#C0C0C0" BORDERCOLORDARK="#808080" FRAME="BOX" RULES="ALL" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP HEIGHT= 401 ><NOBR><div> <IMG SRC="1377af890.gif" border=0 width="431" height="393" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> <tD VALIGN=TOP><NOBR><bR> <bR> <bR> <div> <IMG SRC="6d32cf20.gif" border=0 width="300" height="242" ALT="Imagen de mapa de bits" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> </tR> </TABLE></div> <div> <IMG SRC="137af3cf0.gif" border=0 width="755" height="463" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></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> <TABLE BORDER="0" CELLPADDING="2" CELLSPACING="1" WIDTH="803" BORDERCOLORLIGHT="#C0C0C0" BORDERCOLORDARK="#808080" FRAME="BOX" RULES="ALL" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP HEIGHT= 431 ><NOBR><div> <IMG SRC="137bafa70.gif" border=0 width="431" height="423" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> <tD VALIGN=TOP><NOBR><bR> <bR> <bR> <div> <IMG SRC="6d62c0e0.gif" border=0 width="300" height="270" ALT="Imagen de mapa de bits" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> </tR> </TABLE></div> <div> <IMG SRC="137c18410.gif" border=0 width="792" height="577" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></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> <TABLE BORDER="0" CELLPADDING="2" CELLSPACING="1" WIDTH="803" BORDERCOLORLIGHT="#C0C0C0" BORDERCOLORDARK="#808080" FRAME="BOX" RULES="ALL" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP HEIGHT= 401 ><NOBR><div> <IMG SRC="137eaf890.gif" border=0 width="431" height="393" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> <tD VALIGN=TOP><NOBR><bR> <bR> <bR> <div> <IMG SRC="6d953180.gif" border=0 width="339" height="280" ALT="Imagen de mapa de bits" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> </tR> </TABLE></div> <div> <IMG SRC="138058140.gif" border=0 width="856" height="276" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></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> <TABLE BORDER="0" CELLPADDING="2" CELLSPACING="1" WIDTH="803" BORDERCOLORLIGHT="#C0C0C0" BORDERCOLORDARK="#808080" FRAME="BOX" RULES="ALL" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP HEIGHT= 401 ><NOBR><div> <IMG SRC="1381af890.gif" border=0 width="431" height="393" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> <tD VALIGN=TOP><NOBR><bR> <bR> <bR> <div> <IMG SRC="6dc26120.gif" border=0 width="294" height="274" ALT="Imagen de mapa de bits" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> </tR> </TABLE></div> <div> <IMG SRC="138358b80.gif" border=0 width="856" height="440" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></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> <TABLE BORDER="0" CELLPADDING="2" CELLSPACING="1" WIDTH="803" BORDERCOLORLIGHT="#C0C0C0" BORDERCOLORDARK="#808080" FRAME="BOX" RULES="ALL" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP HEIGHT= 399 ><NOBR><div> <IMG SRC="13848c870.gif" border=0 width="396" height="391" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> <tD VALIGN=TOP><NOBR><bR> <bR> <bR> <div> <IMG SRC="6df271e0.gif" border=0 width="295" height="286" ALT="Imagen de mapa de bits" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> </tR> </TABLE></div> <div> <IMG SRC="138a57220.gif" border=0 width="855" height="546" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="15"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>15</div> <div> <IMG SRC="138b10310.gif" border=0 width="784" height="49" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <IMG SRC="138970cc0.gif" border=0 width="880" height="972" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="16"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>16</div> <div> <IMG SRC="138e84310.gif" border=0 width="644" height="49" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <IMG SRC="138d70970.gif" border=0 width="880" height="919" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="ecuaciones_en_las_que_falta_y"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>Ecuaciones en las que falta y</div> <div> <IMG SRC="139148870.gif" border=0 width="840" height="135" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <FONT SIZE="4" COLOR="#0099FF" FACE="Bell MT" ><B>E</B></FONT>n cada uno de los problemas <FONT SIZE="4" COLOR="#CC0033" FACE="Bell MT" >17</FONT> a <FONT SIZE="4" COLOR="#CC0033" FACE="Bell MT" >22</FONT>, aplique esta sustitución para resolver la ecuación dada.</div> <div>  <A NAME="17"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>17</div> <div> <IMG SRC="13930bcc0.gif" border=0 width="523" height="460" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="18"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>18</div> <div> <IMG SRC="1395ede70.gif" border=0 width="493" height="487" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="19"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>19</div> <div> <IMG SRC="14f15cc90.gif" border=0 width="860" height="457" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="20"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>20</div> <div> <IMG SRC="14f588670.gif" border=0 width="904" height="1127" 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> <bR> </td> <td valign="top" width=105 BGCOLOR="#003366" TEXT="#F7F7FF"> <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="id138_m.htm#ecuación_diferencial_de_segundo_orden" TARGET="TRLX_Middle" TITLE="3.1"><U><IMG BORDER="0" SRC="Wing0a7300a800f0ffffcc00.gif" HEIGHT="16" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Ecuación diferencial de segundo orden</U></A></div> <div> <A HREF="id138_m.htm#ed_lineal_de_segundo_orden" TARGET="TRLX_Middle" TITLE="3.1"><U><IMG BORDER="0" SRC="Wing0a7300a800f0ffffcc00.gif" HEIGHT="16" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Ecuación diferencial lineal de segundo orden</U></A></div> <div> <A HREF="id138_m.htm#problema_con_valor_inicial" TARGET="TRLX_Middle" TITLE="3.1"><U><IMG BORDER="0" SRC="Wing0a7300a800f0ffffcc00.gif" HEIGHT="16" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Problema con valor inicial</U></A></div> <div> <A HREF="id138_m.htm#ecuación_homogenea" TARGET="TRLX_Middle" TITLE="3.1"><U><IMG BORDER="0" SRC="Wing0a7300a800f0ffffcc00.gif" HEIGHT="16" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Ecuación diferencial lineal homogénea de segundo orden</U></A></div> <div> <A HREF="id138_m.htm#ecuación_característica" TARGET="TRLX_Middle" TITLE="3.1"><U><IMG BORDER="0" SRC="Wing0a7300a800f0ffffcc00.gif" HEIGHT="16" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Ecuación carcterística</U></A></div> <div> <A HREF="id138_m.htm#solución_general" TARGET="TRLX_Middle" TITLE="3.1"><U><IMG BORDER="0" SRC="Wing0a7300a800f0ffffcc00.gif" HEIGHT="16" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Solución general de la ED lineal homogénea de segundo orden con coeficientes constantes</U></A></div> <div> <A HREF="id138_m.htm#enunciados" TARGET="TRLX_Middle" TITLE="3.1"><U><IMG BORDER="0" SRC="Wing0a7300a800f0ffffcc00.gif" HEIGHT="16" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Enunciados de los problemas</U></A></div> <div> <A HREF="id138_m.htm#soluciones_1" TARGET="TRLX_Middle" TITLE="3.1"><U><IMG BORDER="0" SRC="Wing0a7300a800f0ffffcc00.gif" HEIGHT="16" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Soluciones de  los problemas</U></A></div> </td> </tr></table></body>