2.2

<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="Bell MT" ><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="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><A HREF="id225.htm" TARGET="_top" TITLE="Ecuaciones lineales de primer orden"><U><B>Apuntes de ecuaciones lineales de primer orden</B></U></A></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>   |   <B>2.2</B>   |   <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>   |   <A HREF="id138.htm" TARGET="_top" TITLE="3.1"><U>3.1</U></A>   |   <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" 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 colspan=2 BGCOLOR="#FFFFFF" TEXT="#080000"> <div> <B>Problemas </B><B>2.2</B></div> <div> <B>Ecuaciones diferenciales de primer orden: otras consideraciones de las ecuaciones lineales.</B></div> <div> <B>Ecuaciones de Bernoulli</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>Enunciados</B></div> <div> .</div> <div> <FONT SIZE="4" COLOR="#009900" FACE="Bell MT" ><B>E</B></FONT>n cada uno de los problemas <FONT SIZE="4" COLOR="#CC0033" FACE="Bell MT" >1</FONT> a <FONT SIZE="4" COLOR="#CC0033" FACE="Bell MT" >4</FONT>, halle la solución general de la ecuación diferencial dada:</div> <div> <FONT SIZE="4" COLOR="#000000" FACE="Bell MT" ><IMG SRC="d7ad4170.gif" border=0 width="212" height="23" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </FONT><A HREF="#1" TITLE="2.2"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="d7cf0180.gif" border=0 width="240" height="24" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#2" TITLE="2.2"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="da74c170.gif" border=0 width="332" height="23" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#3" TITLE="2.2"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <div> <IMG SRC="da8a9180.gif" border=0 width="169" height="24" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#4" TITLE="2.2"><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="#009900" FACE="Bell MT" ><B>E</B></FONT>n cada uno de los problemas <FONT SIZE="4" COLOR="#CC0033" FACE="Bell MT" >5</FONT> al <FONT SIZE="4" COLOR="#CC0033" FACE="Bell MT" >12</FONT>, determine la solución del problema con valor inicial dado. Escriba el intervalo en que la solución es válida:</div> <div> <IMG SRC="db2f5280.gif" border=0 width="245" height="40" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#5" TITLE="2.2"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="db3ad180.gif" border=0 width="173" height="24" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#6" TITLE="2.2"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="db413170.gif" border=0 width="275" height="23" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#7" TITLE="2.2"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A><IMG SRC="db5ef170.gif" border=0 width="239" height="23" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#8" TITLE="2.2"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="db620170.gif" border=0 width="288" height="23" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#9" TITLE="2.2"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <div> <IMG SRC="db74f180.gif" border=0 width="335" height="24" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#10" TITLE="2.2"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="db8f3180.gif" border=0 width="243" height="24" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#11" TITLE="2.2"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="db925180.gif" border=0 width="293" height="24" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#12" TITLE="2.2"><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="#009900" FACE="Bell MT" ><B>C</B></FONT>ada una de las ecuaciones de los problemas del <FONT SIZE="4" COLOR="#CC0033" FACE="Bell MT" >13</FONT> al <FONT SIZE="4" COLOR="#CC0033" FACE="Bell MT" >16</FONT> tiene por lo menos un coeficiente discontinuo en  <I>x</I> = 0. Resuelva cada ecuación para  <I>x</I> > 0  y describa el comportamiento de la solución cuando <I>x</I> tiende a 0, para varios valores de la constante de integración. Trace varios miembros de la familia de curvas integrales:</div> <div> <FONT SIZE="4" COLOR="#000000" FACE="Bell MT" ><IMG SRC="db1a7190.gif" border=0 width="167" height="25" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </FONT><A HREF="#13" TITLE="2.2"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" >        <IMG SRC="dba99190.gif" border=0 width="153" height="25" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </FONT><A HREF="#15" TITLE="2.2"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="dbbbb170.gif" border=0 width="187" height="23" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#16" TITLE="2.2"><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="#009900" 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" >20</FONT>, determine (sin resolver el problema) un intervalo en el que se tenga la certeza de que la solución del problema con valor inicial dado no existe:</div> <div> <FONT SIZE="4" COLOR="#000000" FACE="Bell MT" ><IMG SRC="dbc04170.gif" border=0 width="260" height="23" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </FONT><A HREF="#17" TITLE="2.2"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" >       <IMG SRC="dbdfb170.gif" border=0 width="251" height="23" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </FONT><A HREF="#18" TITLE="2.2"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="dbe0f190.gif" border=0 width="271" height="25" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#19" TITLE="2.2"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <div> <IMG SRC="dbfdc150.gif" border=0 width="220" height="21" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#20" TITLE="2.2"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <div> <IMG SRC="dc039300.gif" border=0 width="825" height="48" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#21" TITLE="2.2"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <div> <IMG SRC="dc154430.gif" border=0 width="852" height="67" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#22" TITLE="2.2"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <div>  <A NAME="ecuaciones_de_bernoulli"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>Ecuaciones de Bernoulli</div> <div> <B>Ecuaciones de Bernoulli</B>.<FONT SIZE="4" COLOR="#000000" FACE="Bell MT" >  Algunas veces es posible resolver una ecuación no lineal al realizar un cambio de la variable dependiente que la convierta en una ecuación lineal. La clase más importante de estas ecuaciones es de la forma</FONT></div> <div> <IMG SRC="d9db3190.gif" border=0 width="179" height="25" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> Estas ecuaciones se conocen como ecuaciones de Bernoulli, en honor a Jacob Bernoulli.</div> <div> <FONT SIZE="4" COLOR="#009900" FACE="Bell MT" ><B>L</B></FONT>os problemas <FONT SIZE="4" COLOR="#CC0033" FACE="Bell MT" >27</FONT> a <FONT SIZE="4" COLOR="#CC0033" FACE="Bell MT" >31</FONT> tratan de las ecuaciones de este tipo:</div> <div> <IMG SRC="dc2814d0.gif" border=0 width="897" height="77" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <FONT SIZE="4" COLOR="#003333" FACE="Bell MT" ><A HREF="#27" TITLE="2.2"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></FONT></div> <div> <FONT SIZE="4" COLOR="#009900" FACE="Bell MT" ><B>E</B></FONT>n cada uno de los problemas <FONT SIZE="4" COLOR="#CC0033" FACE="Bell MT" >28</FONT> a <FONT SIZE="4" COLOR="#CC0033" FACE="Bell MT" >31</FONT>, la ecuación dada es de Bernoulli. En cada caso, resuélvala aplicando la sustitución mencionada en el problema <FONT SIZE="4" COLOR="#CC0033" FACE="Bell MT" >27 b)</FONT><FONT SIZE="4" COLOR="#111111" FACE="Bell MT" >:</FONT></div> <div> <FONT SIZE="4" COLOR="#111111" FACE="Bell MT" ><IMG SRC="dc3db190.gif" border=0 width="219" height="25" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </FONT><A HREF="#28" TITLE="2.2"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A> </div> <div> <IMG SRC="dc500190.gif" border=0 width="768" height="25" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#29" TITLE="2.2"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <div> <IMG SRC="dc673190.gif" border=0 width="883" height="25" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#30" TITLE="2.2"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <div> <IMG SRC="dc73c310.gif" border=0 width="828" height="49" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#31" TITLE="2.2"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <bR> <DIV ALIGN="LEFT"></DIV> <div> <B>Soluciones</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="d797cf00.gif" border=0 width="636" height="240" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <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="d7b556f0.gif" border=0 width="597" height="367" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <IMG SRC="12362c6b0.gif" border=0 width="812" height="363" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <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="d7d3f380.gif" border=0 width="831" height="312" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <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="d7e39f10.gif" border=0 width="569" height="241" 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" >5</FONT> al <FONT SIZE="4" COLOR="#CC0033" FACE="Bell MT" >12</FONT>, determine la solución del problema con valor inicial dado. Escriba el intervalo en que la solución es válida:</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="d7f79b50.gif" border=0 width="889" height="181" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <TABLE BORDER="0" CELLPADDING="1" CELLSPACING="1" WIDTH="684" BORDERCOLORLIGHT="#C0C0C0" BORDERCOLORDARK="#808080" FRAME="BOX" RULES="ALL" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP HEIGHT= 396 ><NOBR><div> <IMG SRC="d809d840.gif" border=0 width="413" height="388" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> <tD VALIGN=TOP WIDTH="304"><bR> <bR> <bR> <bR> <bR> </tD> </tR> </TABLE></div> <bR> <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="66ead180.gif" border=0 width="173" height="24" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <IMG SRC="d81e0a10.gif" border=0 width="736" height="417" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <DIV ALIGN="LEFT"></DIV> <div> <IMG SRC="671391c0.gif" border=0 width="313" height="284" ALT="Imagen de mapa de bits" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <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="d82746f0.gif" border=0 width="884" height="623" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <DIV ALIGN="LEFT"></DIV> <div> <IMG SRC="675371b0.gif" border=0 width="311" height="283" ALT="Imagen de mapa de bits" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <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="d8397c40.gif" border=0 width="919" height="708" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <DIV ALIGN="LEFT"></DIV> <div> <IMG SRC="6798bfd0.gif" border=0 width="395" height="253" ALT="Imagen de mapa de bits" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <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="d84578b0.gif" border=0 width="855" height="651" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <DIV ALIGN="LEFT"></DIV> <div> <IMG SRC="67d47190.gif" border=0 width="327" height="281" ALT="Imagen de mapa de bits" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="10"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>10</div> <div> <IMG SRC="d86df740.gif" border=0 width="735" height="628" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <DIV ALIGN="LEFT"></DIV> <div> <IMG SRC="d87f9330.gif" border=0 width="249" height="307" ALT="Imagen de mapa de bits" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="11"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>11</div> <div> <IMG SRC="d89e3700.gif" border=0 width="739" height="624" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <DIV ALIGN="LEFT"></DIV> <div> <IMG SRC="686a6210.gif" border=0 width="422" height="289" ALT="Imagen de mapa de bits" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="12"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>12</div> <div> <IMG SRC="d8bf0cc0.gif" border=0 width="752" height="716" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <DIV ALIGN="LEFT"></DIV> <div> <IMG SRC="689fe130.gif" border=0 width="254" height="275" ALT="Imagen de mapa de bits" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div> <FONT SIZE="4" COLOR="#0099FF" FACE="Bell MT" ><B>C</B></FONT>ada una de las ecuaciones de los problemas del <FONT SIZE="4" COLOR="#CC0033" FACE="Bell MT" >13</FONT> al <FONT SIZE="4" COLOR="#CC0033" FACE="Bell MT" >16</FONT> tiene por lo menos un coeficiente discontinuo en  <I>x</I> = 0. Resuelva cada ecuación para  <I>x</I> > 0  y describa el comportamiento de la solución cuando <I>x</I> tiende a 0, para varios valores de la constante de integración. Trace varios miembros de la familia de curvas integrales:</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> <IMG SRC="d8d718b0.gif" border=0 width="881" height="651" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <DIV ALIGN="LEFT"></DIV> <div> <IMG SRC="68f331a0.gif" border=0 width="307" height="282" ALT="Imagen de mapa de bits" ALIGN="BOTTOM" HSPACE="0" VSPACE="0">       <IMG SRC="68e10f00.gif" border=0 width="272" height="240" ALT="Imagen de mapa de bits" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <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="d9069ab0.gif" border=0 width="873" height="427" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <DIV ALIGN="LEFT"></DIV> <div> <IMG SRC="69520040.gif" border=0 width="288" height="260" ALT="Imagen de mapa de bits" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <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="d92bfdc0.gif" border=0 width="959" height="476" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <DIV ALIGN="LEFT"></DIV> <div> <IMG SRC="694664c0.gif" border=0 width="358" height="332" ALT="Imagen de mapa de bits" 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" >17</FONT> a <FONT SIZE="4" COLOR="#CC0033" FACE="Bell MT" >20</FONT>, determine (sin resolver el problema) un intervalo en el que se tenga la certeza de que la solución del problema con valor inicial dado no existe.</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="d9454ab0.gif" border=0 width="852" height="171" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <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="d9588b30.gif" border=0 width="648" height="179" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <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="d966ba10.gif" border=0 width="619" height="161" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <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="d97698d0.gif" border=0 width="617" height="141" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="21"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>21</div> <div> <IMG SRC="d9939800.gif" border=0 width="825" height="384" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="22"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>22</div> <div> <IMG SRC="d9b54e50.gif" border=0 width="852" height="485" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div> <B>Ecuaciones de Bernoulli</B>.<FONT SIZE="4" COLOR="#000000" FACE="Bell MT" >  Algunas veces es posible resolver una ecuación no lineal al realizar un cambio de la variable dependiente que la convierta en una ecuación lineal. La clase más importante de estas ecuaciones es de la forma</FONT></div> <div> <IMG SRC="d9cb3190.gif" border=0 width="179" height="25" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> Estas ecuaciones se conocen como ecuaciones de Bernoulli, en honor a Jacob Bernoulli.</div> <div> <FONT SIZE="4" COLOR="#0099FF" FACE="Bell MT" ><B>L</B></FONT>os problemas <FONT SIZE="4" COLOR="#CC0033" FACE="Bell MT" >27</FONT> a <FONT SIZE="4" COLOR="#CC0033" FACE="Bell MT" >31</FONT> tratan de las ecuaciones de este tipo:</div> <div>  <A NAME="27"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>27</div> <div> <IMG SRC="d9f81350.gif" border=0 width="897" height="821" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div> <FONT SIZE="4" COLOR="#00CC00" FACE="Bell MT" ><B>E</B></FONT>n cada uno de los problemas <FONT SIZE="4" COLOR="#CC0033" FACE="Bell MT" >28</FONT> a <FONT SIZE="4" COLOR="#CC0033" FACE="Bell MT" >31</FONT>, la ecuación dada es de Bernoulli. En cada caso, resuélvala aplicando la sustitución mencionada en el problema <FONT SIZE="4" COLOR="#CC0033" FACE="Bell MT" >27 b)</FONT>.</div> <div>  <A NAME="28"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>28</div> <div> <IMG SRC="da073780.gif" border=0 width="627" height="632" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="29"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>29</div> <div> <IMG SRC="da200690.gif" border=0 width="768" height="617" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="30"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>30</div> <div> <IMG SRC="da4737f0.gif" border=0 width="883" height="639" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="31"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>31</div> <div> <IMG SRC="da63c040.gif" border=0 width="828" height="772" 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> </tr><tr> <td valign="top" colspan=2 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="100" /> <input type="submit" name="sa" value="Buscar" /> </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>