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="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><FONT SIZE="4" COLOR="#FFFFFF" FACE="Bell MT" ><B>|</B></FONT><FONT SIZE="3" COLOR="#FFFFFF" FACE="Georgia" > <A HREF="id84.htm" TARGET="_top" TITLE="G. F. Simmons"><U>G. F. Simmons</U></A></FONT><FONT SIZE="4" COLOR="#FFFFFF" FACE="Georgia" > |</FONT></div> <div> <B>1.2</B>   |   <A HREF="id125.htm" TARGET="_top" TITLE="2.7"><U>2.7</U></A>   |   <A HREF="id280.htm" TARGET="_top" TITLE="2.10"><U>2.10</U></A>   |   <A HREF="id126.htm" TARGET="_top" TITLE="3.14"><U>3.14</U></A>   |   <A HREF="id127.htm" TARGET="_top" TITLE="3.15"><U>3.15</U></A>   |   <A HREF="id128.htm" TARGET="_top" TITLE="3.16"><U>3.16</U></A>   |   <A HREF="id232.htm" TARGET="_top" TITLE="3.17"><U>3.17</U></A>   |   <A HREF="id290.htm" TARGET="_top" TITLE="Created from: 3.17"><U>Created from: 3.17</U></A>   |   <A HREF="id237.htm" TARGET="_top" TITLE="3.18"><U>3.18</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>Proble</B><B>ma</B><B>s </B><B>1.2</B></div> <div> <B>Generalidades sobre las soluciones </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> <bR> <div> <B>1.</B><FONT SIZE="4" COLOR="#00CC00" FACE="Bell MT" ><B>  </B></FONT><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" >Verificar que las siguientes funciones (explícitas o implícitas) son soluciones de las correspondientes ecuaciones diferenciales:</FONT></div> <div> <FONT SIZE="4" COLOR="#000000" FACE="Bell MT" ><IMG SRC="e29b5190.gif" border=0 width="181" height="25" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </FONT><A HREF="#1_a" TITLE="1.2"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="e2bd11b0.gif" border=0 width="209" height="27" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#1_c" TITLE="1.2"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="e2d2d180.gif" border=0 width="301" height="24" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#1_e" TITLE="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="e2f45180.gif" border=0 width="325" height="24" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </FONT><A HREF="#1_g" TITLE="1.2"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="e31291f0.gif" border=0 width="297" height="31" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="http://youtu.be/8-PfWGZ9Vv0" TARGET="_top" TITLE="http://youtu.be/8-PfWGZ9Vv0"><U><IMG SRC="c4218180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A> </div> <div> <IMG SRC="e33ef2f0.gif" border=0 width="239" height="47" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#1_j" TITLE="1.2"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="e3519190.gif" border=0 width="281" height="25" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#1_o" TITLE="1.2"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <div> <U>Nota</U>: para verificar que una función es la solución de una ecuación diferencial dada es menester hallar, dependiendo del orden de la ED, la primera, la segunda, etc. derivadas de la función (si la ED es de primer orden sólo es necesario hallar la primera derivada, si es de segundo orden hay que hallar tanto la primera como la segunda derivada de la función, etc.) luego sustituir en la ecuación diferencial estas funciones y constatar que se obtiene una identidad.</div> <div> <B>2.</B><FONT SIZE="4" COLOR="#000000" FACE="Georgia" > </FONT><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" > Hallar la solución general de cada una de las ecuaciones diferenciales siguientes:</FONT></div> <div> <FONT SIZE="4" COLOR="#000000" FACE="Bell MT" ><IMG SRC="e37741b0.gif" border=0 width="116" height="27" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </FONT><A HREF="#2_a" TITLE="1.2"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="e39641d0.gif" border=0 width="100" height="29" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#2_c" TITLE="1.2"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>      <IMG SRC="e3b78170.gif" border=0 width="120" height="23" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#2_e" TITLE="1.2"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>      <IMG SRC="e3d85190.gif" border=0 width="133" height="25" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#2_f" TITLE="1.2"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="e3f83190.gif" border=0 width="131" height="25" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#2_g" TITLE="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="e417d190.gif" border=0 width="125" height="25" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#2_m" TITLE="1.2"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="e43a1170.gif" border=0 width="161" height="23" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#2_r" TITLE="1.2"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <div> <B>3.</B><FONT SIZE="4" COLOR="#000000" FACE="Georgia" >  </FONT><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" >Para cada una de las ecuaciones diferenciales siguientes, hallar la solución particular que satisface la condición inicial dada:</FONT></div> <div> <FONT SIZE="4" COLOR="#000000" FACE="Bell MT" ><IMG SRC="e4507190.gif" border=0 width="263" height="25" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </FONT><A HREF="#3_a" TITLE="1.2"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="e472b190.gif" border=0 width="299" height="25" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#3_d" TITLE="1.2"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <div> <B>4. </B><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" > Para cada una de las siguientes ecuaciones diferenciales, hallar la curva integral que pasa por el punto dado:</FONT></div> <div> <FONT SIZE="4" COLOR="#000000" FACE="Bell MT" ><IMG SRC="e49a4190.gif" border=0 width="164" height="25" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </FONT><A HREF="#4_a" TITLE="1.2"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="e4baf170.gif" border=0 width="431" height="23" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#4_d" TITLE="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="e4de3290.gif" border=0 width="483" height="41" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#5" TITLE="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="e4f6d690.gif" border=0 width="877" height="105" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#6" TITLE="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="e5168190.gif" border=0 width="872" height="25" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#8" TITLE="1.2"><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_a"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>1_a</div> <div> <IMG SRC="e28d59b0.gif" border=0 width="725" height="155" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="1_c"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>1_c</div> <div> <IMG SRC="e2ad5ec0.gif" border=0 width="725" height="236" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="1_e"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>1_e</div> <div> <IMG SRC="e2cafec0.gif" border=0 width="687" height="236" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="1_g"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>1_g</div> <div> <IMG SRC="e3001d70.gif" border=0 width="769" height="215" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="1_j"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>1_j</div> <div> <IMG SRC="e3201270.gif" border=0 width="769" height="295" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="1_o"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>1_o</div> <div> <IMG SRC="e3478340.gif" border=0 width="632" height="308" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div> <B>2.</B><FONT SIZE="4" COLOR="#000000" FACE="Georgia" > </FONT><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" > Hallar la solución general de cada una de las ecuaciones diferenciales siguientes:</FONT></div> <div>  <A NAME="2_a"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>2_a</div> <div> <IMG SRC="e36557d0.gif" border=0 width="341" height="125" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="2_c"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>2_c</div> <div> <IMG SRC="e38a5830.gif" border=0 width="421" height="131" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="2_e"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>2_e</div> <div> <IMG SRC="e3aa98c0.gif" border=0 width="681" height="140" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="2_f"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>2_f</div> <div> <IMG SRC="e3cb88f0.gif" border=0 width="440" height="143" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="2_g"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>2_g</div> <div> <IMG SRC="e3ee5cb0.gif" border=0 width="741" height="971" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="2_m"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>2_m</div> <div> <IMG SRC="e407b900.gif" border=0 width="635" height="144" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="2_r"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>2_r</div> <div> <IMG SRC="e42b05b0.gif" border=0 width="688" height="347" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div> <B>3.</B><FONT SIZE="4" COLOR="#000000" FACE="Georgia" >  </FONT><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" >Para cada una de las ecuaciones diferenciales siguientes, hallar la solución particular que satisface la condición inicial dada:</FONT></div> <div>  <A NAME="3_a"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>3_a</div> <div> <IMG SRC="e4478dc0.gif" border=0 width="632" height="476" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="3_d"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>3_d</div> <div> <IMG SRC="e46bb230.gif" border=0 width="699" height="291" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div> <B>4. </B><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" > Para cada una de las siguientes ecuaciones diferenciales, hallar la curva integral que pasa por el punto dado:</FONT></div> <div>  <A NAME="4_a"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>4_a</div> <div> <IMG SRC="e489f050.gif" border=0 width="671" height="261" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="4_d"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>4_d</div> <div> <IMG SRC="e4ad8ff0.gif" border=0 width="728" height="255" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <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="e4cf5f90.gif" border=0 width="501" height="249" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></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="e4e6dfb0.gif" border=0 width="877" height="507" ALT="MathType 6.0 Equation" 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="e506dbc0.gif" border=0 width="877" height="188" 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" 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> <bR> </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 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