4.1

<body topmargin=0 leftmargin=0 marginheight=0 marginwidth=0> <table cellpadding=0 cellspacing=0 border=0><tr> <td valign="top" colspan=2 height=24 BGCOLOR="#003366" TEXT="#F7F7FF"> <div> <FONT SIZE="4" COLOR="#FFFFFF" FACE="Bell MT" ><B>| </B><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="3" COLOR="#FFFFFF" FACE="Georgia" ><B> |  </B><A HREF="id175.htm" TARGET="_top" TITLE="NagleSaffSnider"><U><B>Nagle-Saff-Snider</B></U></A></FONT><FONT SIZE="4" COLOR="#FFFFFF" FACE="Georgia" ><B> </B></FONT></div> <div> <A HREF="id177.htm" TARGET="_top" TITLE="1.2"><U>1.2</U></A>   |   <A HREF="id178.htm" TARGET="_top" TITLE="2.2"><U>2.2</U></A>   |   <A HREF="id179.htm" TARGET="_top" TITLE="2.3"><U>2.3</U></A>   |   <B>4.1</B>   |   <A HREF="id180.htm" TARGET="_top" TITLE="4.2"><U>4.2</U></A>   |   <A HREF="id277.htm" TARGET="_top" TITLE="4.3"><U>4.3</U></A>   |   <A HREF="id278.htm" TARGET="_top" TITLE="4.4"><U>4.4</U></A>   |   <A HREF="id181.htm" TARGET="_top" TITLE="7.2"><U>7.2</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> <bR> </td> <td valign="top" height=456 BGCOLOR="#FFFFFF" TEXT="#080000"> <div> <B>Ejercicios </B><B>4.1</B></div> <div> <B>Ecuaciones lineales de segundo orden. Introducción: El oscilador masa - resorte</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> <B> <A NAME="enunciados"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A></B><B>enunciados</B></div> <div> <B>Enunciados de los ejercicios</B></div> <bR> <bR> <div> <IMG SRC="140c8b370.gif" border=0 width="907" height="55" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <DIV ALIGN="LEFT"></DIV> <div> <A HREF="#1" TITLE="4.1"><U>Ver la solución</U></A><FONT SIZE="4" COLOR="#CC0033" FACE="Cambria" > </FONT><A HREF="#1" TITLE="4.1"><U><IMG SRC="c4118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <bR> <div> <IMG SRC="141279670.gif" border=0 width="889" height="103" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <DIV ALIGN="LEFT"></DIV> <div> <A HREF="#2" TITLE="4.1"><U>Ver la solución</U></A> <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></div> <bR> <div> <IMG SRC="1416f3530.gif" border=0 width="755" height="83" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </div> <DIV ALIGN="LEFT"></DIV> <div> <A HREF="#3" TITLE="4.1"><U>Ver la solución</U></A> <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></div> <bR> <div> <IMG SRC="144074410.gif" border=0 width="884" height="65" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <DIV ALIGN="LEFT"></DIV> <div> <A HREF="#4" TITLE="4.1"><U>Ver la solución</U></A> <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></div> <bR> <div> <IMG SRC="1442744c0.gif" border=0 width="884" height="76" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <DIV ALIGN="LEFT"></DIV> <div> <FONT SIZE="3" COLOR="#000000" FACE="Georgia" ><A HREF="#5" TITLE="4.1"><U>Ver la solución</U></A></FONT> <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></div> <bR> <div> <IMG SRC="144454530.gif" border=0 width="852" height="83" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <DIV ALIGN="LEFT"></DIV> <div> <FONT SIZE="3" COLOR="#000000" FACE="Georgia" ><A HREF="#6" TITLE="4.1"><U>Ver la solución</U></A></FONT> <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></div> <bR> <div> <FONT SIZE="4" COLOR="#339900" FACE="Georgia" ><B>           </B></FONT><FONT SIZE="4" COLOR="#CC0033" FACE="Georgia" ><B>E</B></FONT>n los problemas <FONT SIZE="4" COLOR="#CC0033" FACE="Georgia" >7 </FONT>a <FONT SIZE="4" COLOR="#CC0033" FACE="Georgia" >9</FONT>, determine una solución sincrónica de la forma <IMG SRC="144799180.gif" border=0 width="153" height="24" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> a la ecuación de oscilador forzado dada, hallando los valores numéricos de <I>A</I> y <I>B</I> respectivamente.</div> <bR> <div> <IMG SRC="144804190.gif" border=0 width="260" height="25" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <DIV ALIGN="LEFT"></DIV> <div> <A HREF="#7" TITLE="4.1"><U>Ver la solución</U></A> <FONT SIZE="4" COLOR="#000000" FACE="Georgia" ><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></FONT></div> <div> <IMG SRC="144c04190.gif" border=0 width="260" height="25" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <DIV ALIGN="LEFT"></DIV> <div> <A HREF="#8" TITLE="4.1"><U>Ver la solución</U></A> <FONT SIZE="4" COLOR="#000000" FACE="Georgia" ><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></FONT></div> <div> <IMG SRC="144d4f190.gif" border=0 width="335" height="25" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <DIV ALIGN="LEFT"></DIV> <div> <FONT SIZE="3" COLOR="#000000" FACE="Georgia" ><A HREF="#9" TITLE="4.1"><U>Ver la solución</U></A></FONT> <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></div> <bR> <div> <FONT SIZE="4" COLOR="#CC0033" FACE="Times New Roman" ><B>10.</B></FONT>  Los osciladores no amortiguados forzados en la resonancia tienen soluciones inusuales (no físicas).</div> <div> (<FONT SIZE="4" COLOR="#CC0033" FACE="Georgia" >a</FONT>)<B>  </B>Para estudiar esto, determine la solución sincrónica <IMG SRC="144f99140.gif" border=0 width="153" height="20" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><I> </I>de la ecuación genérica del oscilador forzado</div> <DIV ALIGN="LEFT"></DIV> <div> <IMG SRC="1450b5190.gif" border=0 width="181" height="25" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <IMG SRC="145164590.gif" border=0 width="868" height="89" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <DIV ALIGN="LEFT"></DIV> <div> <A HREF="#10" TITLE="4.1"><U>Ver la solución</U></A> <FONT SIZE="4" COLOR="#000000" FACE="Georgia" ><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></div> <div>  <A NAME="soluciones"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>Soluciones</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="140d8c870.gif" border=0 width="908" height="391" 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="14137cab0.gif" border=0 width="892" height="683" 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="143b49930.gif" border=0 width="841" height="403" 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></div> <div> <IMG SRC="144174290.gif" border=0 width="884" height="553" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0">4</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="144374240.gif" border=0 width="884" height="548" 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="144677300.gif" border=0 width="887" height="816" 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="144a69010.gif" border=0 width="873" height="513" 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="144b69010.gif" border=0 width="873" height="513" 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="144e6bcf0.gif" border=0 width="875" height="463" 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> <IMG SRC="145287550.gif" border=0 width="903" height="853" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> Las gráficas de estas funciones las trace en el programa Geogebra:</div> <DIV ALIGN="LEFT"></DIV> <div> <IMG SRC="14537e6d0.gif" border=0 width="638" height="365" ALT="Imagen de Paintbrush" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <U><IMG SRC="14555f470.gif" border=0 width="607" height="71" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></div> <DIV ALIGN="LEFT"></DIV> <div> <IMG SRC="1454d15d0.gif" border=0 width="465" height="349" ALT="Imagen de Paintbrush" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></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> </td> </tr></table></body>