Análisis de Sistemas de Tiempo Continuo en el Dominio del Tiempo - Sistemas Lineales CT y Ecuaciones Diferenciales

""<DIV class=section id=int> <H2><FONT size=2>Sistemas Lineales de Tiempo-Continuo</FONT></H2> <DIV class=para id=p1><FONT size=2>Físicamente realizable, los sistemas lineales invariantes en el tiempo pueden ser descritos por un conjuto de ecuaciones lineales diferenciales: </FONT></DIV> <TABLE class=figure id=fig1 cellSpacing=0 cellPadding=0 width="50%" align=center border=0> <TBODY> <TR> <TD> <DIV class=inner-figure><FONT size=2><IMG class=media src= http://www.wikilearning.com/imagescc/20469/ctlin.png" alt="image" /> ></FONT></DIV> <DIV class=caption><FONT size=2><SPAN class=caption-before><STRONG>Figure 1: </STRONG></SPAN>Describción gráfica de un sistema básico lineal invariante en el tiempo con una entrada, <SPAN class=mrow id=id7713627><SPAN class=mrow id=id7713643><SPAN class=mrow id=id7713656><EM><SPAN class=mi1>f</SPAN><SPAN class=mo id=id7713659></SPAN></EM><SPAN id=id7713660L>(</SPAN><SPAN id=id7713660M><SPAN class=mi1><EM>t</EM></SPAN></SPAN><SPAN id=id7713660R>)</SPAN> <SCRIPT type=text/javascript> var mrowH = id7713660M.offsetHeight; mrowStretch(id7713660L,'æ','ç','ç','è'); mrowStretch(id7713660R,'ö','÷','÷','ø');</SCRIPT> </SPAN></SPAN></SPAN>y una salida, <SPAN class=mrow id=id7713680><SPAN class=mrow id=id7713647><SPAN class=mrow id=id7713687><EM><SPAN class=mi1>y</SPAN><SPAN class=mo id=id7713690></SPAN></EM><SPAN id=id7713691L>(</SPAN><SPAN id=id7713691M><SPAN class=mi1><EM>t</EM></SPAN></SPAN><SPAN id=id7713691R>)</SPAN> <SCRIPT type=text/javascript> var mrowH = id7713691M.offsetHeight; mrowStretch(id7713691L,'æ','ç','ç','è'); mrowStretch(id7713691R,'ö','÷','÷','ø');</SCRIPT> </SPAN></SPAN></SPAN>. </FONT></DIV></TD></TR></TBODY></TABLE> <DIV class=para id=p2><SPAN class=mrow id=id7713686><SPAN class=mrow id=id7713653><SPAN class=mrow id=id7713727><SPAN class=mrow id=id7713728><SPAN class=mrow id=id7713732> <TABLE class=mfrac style="FONT-SIZE: 75%"> <TBODY> <TR class=mfraca id=aid7713589> <TD class=mfracaa><SPAN class=mrow id=id7713736><FONT size=2><SPAN><SPAN class=mo id=id7713738>d</SPAN></SPAN><SUP><SPAN class=mrow id=id7713740><SPAN class=mi1><EM>n</EM></SPAN></SPAN></SUP></FONT></SPAN></TD></TR> <TR class=mfracb id=bid7713589 style="HEIGHT: 22px"> <TD><SPAN class=mrow id=id7713746><FONT size=2><SPAN class=mo id=id7713747>d</SPAN><EM><SPAN><SPAN class=mrow id=id7713750><SPAN class=mi1>t</SPAN></SPAN></SPAN><SUP><SPAN class=mrow id=id7713753><SPAN class=mi1>n</SPAN></SPAN></SUP></EM></FONT></SPAN></TD></TR></TBODY></TABLE> <SCRIPT type=text/javascript> if ( aid7713589.offsetHeight > bid7713589.offsetHeight ) bid7713589.style.setExpression("height",aid7713589.offsetHeight ); else aid7713589.style.setExpression("height",bid7713589.offsetHeight ); </SCRIPT> <SPAN class=mrow id=id7713756><SPAN class=mrow id=id7713761><FONT size=2><EM><SPAN class=mi1>y</SPAN><SPAN class=mo id=id7713764></SPAN></EM><SPAN id=id7713765L>(</SPAN><SPAN id=id7713765M><SPAN class=mi1><EM>t</EM></SPAN></SPAN><SPAN id=id7713765R>)</SPAN> <SCRIPT type=text/javascript> var mrowH = id7713765M.offsetHeight; mrowStretch(id7713765L,'æ','ç','ç','è'); mrowStretch(id7713765R,'ö','÷','÷','ø');</SCRIPT> </FONT></SPAN></SPAN></SPAN><SPAN class=mo id=id7713768><FONT size=2>+</FONT></SPAN><SPAN class=mrow id=id7713770><FONT size=2><SPAN class=mrow id=id7713771><SPAN><SPAN class=mi1><EM>a</EM></SPAN></SPAN><SUB><SPAN class=mrow id=id7713778><SPAN class=mi1><EM>n</EM></SPAN><SPAN class=mo id=id7713783>−</SPAN><SPAN class=mn>1</SPAN></SPAN></SUB></SPAN><SPAN class=mo id=id7713790></SPAN></FONT><SPAN class=mrow id=id7713793><FONT size=2> </FONT> <TABLE class=mfrac style="FONT-SIZE: 75%"> <TBODY> <TR class=mfraca id=aid7713794> <TD class=mfracaa><SPAN class=mrow id=id7713795><FONT size=2><SPAN><SPAN class=mo id=id7713797>d</SPAN></SPAN><SUP><SPAN class=mrow id=id7713800><SPAN class=mrow id=id7713806><SPAN class=mi1><EM>n</EM></SPAN><SPAN class=mo id=id7713810>−</SPAN><SPAN class=mn>1</SPAN></SPAN></SPAN></SUP></FONT></SPAN></TD></TR> <TR class=mfracb id=bid7713794 style="HEIGHT: 22px"> <TD><SPAN class=mrow id=id7713814><FONT size=2><SPAN class=mo id=id7713815>d</SPAN><SPAN><SPAN class=mrow id=id7713818><SPAN class=mi1><EM>t</EM></SPAN></SPAN></SPAN><SUP><SPAN class=mrow id=id7713821><SPAN class=mrow id=id7713822><SPAN class=mi1><EM>n</EM></SPAN><SPAN class=mo id=id7713826>−</SPAN><SPAN class=mn>1</SPAN></SPAN></SPAN></SUP></FONT></SPAN></TD></TR></TBODY></TABLE> <SCRIPT type=text/javascript> if ( aid7713794.offsetHeight > bid7713794.offsetHeight ) bid7713794.style.setExpression("height",aid7713794.offsetHeight ); else aid7713794.style.setExpression("height",bid7713794.offsetHeight ); </SCRIPT> <SPAN class=mrow id=id7713830><SPAN class=mrow id=id7713831><FONT size=2><EM><SPAN class=mi1>y</SPAN><SPAN class=mo id=id7713834></SPAN></EM><SPAN id=id7713835L>(</SPAN><SPAN id=id7713835M><SPAN class=mi1><EM>t</EM></SPAN></SPAN><SPAN id=id7713835R>)</SPAN> <SCRIPT type=text/javascript> var mrowH = id7713835M.offsetHeight; mrowStretch(id7713835L,'æ','ç','ç','è'); mrowStretch(id7713835R,'ö','÷','÷','ø');</SCRIPT> </FONT></SPAN></SPAN></SPAN></SPAN><FONT size=2><SPAN class=mo id=id7713838>+</SPAN><SPAN class=mi1><EM>…</EM></SPAN><SPAN class=mo id=id7713843>+</SPAN></FONT><SPAN class=mrow id=id7713845><FONT size=2><SPAN class=mrow id=id7713846><SPAN><SPAN class=mi1><EM>a</EM></SPAN></SPAN><SUB><SPAN class=mn>1</SPAN></SUB></SPAN><SPAN class=mo id=id7713856></SPAN></FONT><SPAN class=mrow id=id7713857><FONT size=2> </FONT> <TABLE class=mfrac style="FONT-SIZE: 75%"> <TBODY> <TR class=mfraca id=aid7713858> <TD class=mfracaa><SPAN class=mo id=id7713859><FONT size=2>d</FONT></SPAN></TD></TR> <TR class=mfracb id=bid7713858 style="HEIGHT: 20px"> <TD><SPAN class=mrow id=id7713861><FONT size=2><SPAN class=mo id=id7713862>d</SPAN><SPAN class=mi1><EM>t</EM></SPAN></FONT></SPAN></TD></TR></TBODY></TABLE> <SCRIPT type=text/javascript> if ( aid7713858.offsetHeight > bid7713858.offsetHeight ) bid7713858.style.setExpression("height",aid7713858.offsetHeight ); else aid7713858.style.setExpression("height",bid7713858.offsetHeight ); </SCRIPT> <SPAN class=mrow id=id7713867><SPAN class=mrow id=id7713868><FONT size=2><EM><SPAN class=mi1>y</SPAN><SPAN class=mo id=id7713872></SPAN></EM><SPAN id=id7713873L>(</SPAN><SPAN id=id7713873M><SPAN class=mi1><EM>t</EM></SPAN></SPAN><SPAN id=id7713873R>)</SPAN> <SCRIPT type=text/javascript> var mrowH = id7713873M.offsetHeight; mrowStretch(id7713873L,'æ','ç','ç','è'); mrowStretch(id7713873R,'ö','÷','÷','ø');</SCRIPT> </FONT></SPAN></SPAN></SPAN></SPAN><FONT size=2><SPAN class=mo id=id7713876>+</SPAN><SPAN class=mrow id=id7713878><SPAN class=mrow id=id7713879><SPAN><SPAN class=mi1><EM>a</EM></SPAN></SPAN><SUB><SPAN class=mn>0</SPAN></SUB></SPAN><SPAN class=mo id=id7713889></SPAN><SPAN class=mrow id=id7713890><EM><SPAN class=mi1>y</SPAN><SPAN class=mo id=id7713893></SPAN></EM><SPAN id=id7713894L>(</SPAN><SPAN id=id7713894M><SPAN class=mi1><EM>t</EM></SPAN></SPAN><SPAN id=id7713894R>)</SPAN> <SCRIPT type=text/javascript> var mrowH = id7713894M.offsetHeight; mrowStretch(id7713894L,'æ','ç','ç','è'); mrowStretch(id7713894R,'ö','÷','÷','ø');</SCRIPT> </SPAN></SPAN></FONT></SPAN><SPAN class=mo id=id7713897><FONT size=2>=</FONT></SPAN><SPAN class=mrow id=id7713899><SPAN class=mrow id=id7713900><EM><FONT size=2><SPAN class=mrow id=id7713901><SPAN><SPAN class=mi1>b</SPAN></SPAN><SUB><SPAN class=mi1>m</SPAN></SUB></SPAN><SPAN class=mo id=id7713911></SPAN></FONT></EM><SPAN class=mrow id=id7713912><FONT size=2> </FONT> <TABLE class=mfrac style="FONT-SIZE: 75%"> <TBODY> <TR class=mfraca id=aid7713913> <TD class=mfracaa><SPAN class=mrow id=id7713914><FONT size=2><SPAN><SPAN class=mo id=id7713916>d</SPAN></SPAN><SUP><SPAN class=mrow id=id7713918><SPAN class=mi1><EM>m</EM></SPAN></SPAN></SUP></FONT></SPAN></TD></TR> <TR class=mfracb id=bid7713913 style="HEIGHT: 22px"> <TD><SPAN class=mrow id=id7713923><FONT size=2><SPAN class=mo id=id7713924>d</SPAN><EM><SPAN><SPAN class=mrow id=id7713927><SPAN class=mi1>t</SPAN></SPAN></SPAN><SUP><SPAN class=mrow id=id7713930><SPAN class=mi1>m</SPAN></SPAN></SUP></EM></FONT></SPAN></TD></TR></TBODY></TABLE> <SCRIPT type=text/javascript> if ( aid7713913.offsetHeight > bid7713913.offsetHeight ) bid7713913.style.setExpression("height",aid7713913.offsetHeight ); else aid7713913.style.setExpression("height",bid7713913.offsetHeight ); </SCRIPT> <SPAN class=mrow id=id7713933><SPAN class=mrow id=id7713934><FONT size=2><EM><SPAN class=mi1>f</SPAN><SPAN class=mo id=id7713938></SPAN></EM><SPAN id=id7713939L>(</SPAN><SPAN id=id7713939M><SPAN class=mi1><EM>t</EM></SPAN></SPAN><SPAN id=id7713939R>)</SPAN> <SCRIPT type=text/javascript> var mrowH = id7713939M.offsetHeight; mrowStretch(id7713939L,'æ','ç','ç','è'); mrowStretch(id7713939R,'ö','÷','÷','ø');</SCRIPT> </FONT></SPAN></SPAN></SPAN></SPAN><FONT size=2><SPAN class=mo id=id7713942>+</SPAN><SPAN class=mi1><EM>…</EM></SPAN><SPAN class=mo id=id7713946>+</SPAN></FONT><SPAN class=mrow id=id7713948><FONT size=2><SPAN class=mrow id=id7713949><SPAN><SPAN class=mi1><EM>b</EM></SPAN></SPAN><SUB><SPAN class=mn>1</SPAN></SUB></SPAN><SPAN class=mo id=id7713959></SPAN></FONT><SPAN class=mrow id=id7713960><FONT size=2> </FONT> <TABLE class=mfrac style="FONT-SIZE: 75%"> <TBODY> <TR class=mfraca id=aid7713961> <TD class=mfracaa><SPAN class=mo id=id7713962><FONT size=2>d</FONT></SPAN></TD></TR> <TR class=mfracb id=bid7713961 style="HEIGHT: 20px"> <TD><SPAN class=mrow id=id7713964><FONT size=2><SPAN class=mo id=id7713965>d</SPAN><SPAN class=mi1><EM>t</EM></SPAN></FONT></SPAN></TD></TR></TBODY></TABLE> <SCRIPT type=text/javascript> if ( aid7713961.offsetHeight > bid7713961.offsetHeight ) bid7713961.style.setExpression("height",aid7713961.offsetHeight ); else aid7713961.style.setExpression("height",bid7713961.offsetHeight ); </SCRIPT> <SPAN class=mrow id=id7713971><SPAN class=mrow id=id7713972><FONT size=2><EM><SPAN class=mi1>f</SPAN><SPAN class=mo id=id7713975></SPAN></EM><SPAN id=id7713976L>(</SPAN><SPAN id=id7713976M><SPAN class=mi1><EM>t</EM></SPAN></SPAN><SPAN id=id7713976R>)</SPAN> <SCRIPT type=text/javascript> var mrowH = id7713976M.offsetHeight; mrowStretch(id7713976L,'æ','ç','ç','è'); mrowStretch(id7713976R,'ö','÷','÷','ø');</SCRIPT> </FONT></SPAN></SPAN></SPAN></SPAN><FONT size=2><SPAN class=mo id=id7713979>+</SPAN><SPAN class=mrow id=id7713981><SPAN class=mrow id=id7713982><SPAN><SPAN class=mi1><EM>b</EM></SPAN></SPAN><SUB><SPAN class=mn>0</SPAN></SUB></SPAN><SPAN class=mo id=id7713992></SPAN><SPAN class=mrow id=id7713993><EM><SPAN class=mi1>f</SPAN><SPAN class=mo id=id7713996></SPAN></EM><SPAN id=id7713997L>(</SPAN><SPAN id=id7713997M><SPAN class=mi1><EM>t</EM></SPAN></SPAN><SPAN id=id7713997R>)</SPAN> <SCRIPT type=text/javascript> var mrowH = id7713997M.offsetHeight; mrowStretch(id7713997L,'æ','ç','ç','è'); mrowStretch(id7713997R,'ö','÷','÷','ø');</SCRIPT> </SPAN></SPAN></FONT></SPAN></SPAN></SPAN></SPAN><FONT size=2>Equivalentemente, </FONT> <DIV class=equation id=eq1><SPAN class=mrow id=id7714339><SPAN class=mrow id=id7713803><SPAN class=mrow id=id7714341><SPAN class=mrow id=id7714342> <TABLE class=munderover> <TBODY> <TR> <TD><SPAN class=mrow id=id7714358><SPAN class=mi1><EM><FONT size=2>n</FONT></EM></SPAN></SPAN></TD></TR> <TR> <TD><SPAN class=mo id=id7714345><FONT size=2>∑</FONT></SPAN></TD></TR> <TR> <TD><SPAN class=mrow id=id7714347><FONT size=2><SPAN class=mi1><EM>i</EM></SPAN><SPAN class=mo id=id7714352>=</SPAN><SPAN class=mn>0</SPAN></FONT></SPAN></TD></TR></TBODY></TABLE><SPAN id=id7714363L> <TABLE class=lr> <TBODY> <TR> <TD><FONT size=2><FONT face=symbol>æ</FONT><BR><FONT face=symbol>ç</FONT><BR><FONT face=symbol>è</FONT></FONT></TD></TR></TBODY></TABLE></SPAN><SPAN id=id7714363M><SPAN class=mrow id=id7714366><EM><FONT size=2><SPAN class=mrow id=id7714367><SPAN><SPAN class=mi1>a</SPAN></SPAN><SUB><SPAN class=mi1>i</SPAN></SUB></SPAN><SPAN class=mo id=id7714378></SPAN></FONT></EM><SPAN class=mrow id=id7714386><FONT size=2> </FONT> <TABLE class=mfrac style="FONT-SIZE: 75%"> <TBODY> <TR class=mfraca id=aid7714387> <TD class=mfracaa><SPAN class=mrow id=id7714388><FONT size=2><SPAN><SPAN class=mo id=id7714390>d</SPAN></SPAN><SUP><SPAN class=mrow id=id7714392><SPAN class=mi1><EM>i</EM></SPAN></SPAN></SUP></FONT></SPAN></TD></TR> <TR class=mfracb id=bid7714387 style="HEIGHT: 22px"> <TD><SPAN class=mrow id=id7714398><FONT size=2><SPAN class=mo id=id7714399>d</SPAN><EM><SPAN><SPAN class=mrow id=id7714402><SPAN class=mi1>t</SPAN></SPAN></SPAN><SUP><SPAN class=mrow id=id7714407><SPAN class=mi1>i</SPAN></SPAN></SUP></EM></FONT></SPAN></TD></TR></TBODY></TABLE> <SCRIPT type=text/javascript> if ( aid7714387.offsetHeight > bid7714387.offsetHeight ) bid7714387.style.setExpression("height",aid7714387.offsetHeight ); else aid7714387.style.setExpression("height",bid7714387.offsetHeight ); </SCRIPT> <SPAN class=mrow id=id7714411><SPAN class=mrow id=id7714417><FONT size=2><EM><SPAN class=mi1>y</SPAN><SPAN class=mo id=id7714420></SPAN></EM><SPAN id=id7714422L>(</SPAN><SPAN id=id7714422M><SPAN class=mi1><EM>t</EM></SPAN></SPAN><SPAN id=id7714422R>)</SPAN> <SCRIPT type=text/javascript> var mrowH = id7714422M.offsetHeight; mrowStretch(id7714422L,'æ','ç','ç','è'); mrowStretch(id7714422R,'ö','÷','÷','ø');</SCRIPT> </FONT></SPAN></SPAN></SPAN></SPAN></SPAN><SPAN id=id7714363R> <TABLE class=lr> <TBODY> <TR> <TD><FONT size=2><FONT face=symbol>ö</FONT><BR><FONT face=symbol>÷</FONT><BR><FONT face=symbol>ø</FONT></FONT></TD></TR></TBODY></TABLE></SPAN> <SCRIPT type=text/javascript> var mrowH = id7714363M.offsetHeight; mrowStretch(id7714363L,'æ','ç','ç','è'); mrowStretch(id7714363R,'ö','÷','÷','ø');</SCRIPT> </SPAN></SPAN><SPAN class=mrow id=id7714425><SPAN class=mo id=id7714426><FONT size=2>=</FONT></SPAN></SPAN><SPAN class=mrow id=id7714428><SPAN class=mrow id=id7714430><FONT size=2> </FONT> <TABLE class=munderover> <TBODY> <TR> <TD><SPAN class=mrow id=id7714445><SPAN class=mi1><EM><FONT size=2>m</FONT></EM></SPAN></SPAN></TD></TR> <TR> <TD><SPAN class=mo id=id7714432><FONT size=2>∑</FONT></SPAN></TD></TR> <TR> <TD><SPAN class=mrow id=id7714435><FONT size=2><SPAN class=mi1><EM>i</EM></SPAN><SPAN class=mo id=id7714440>=</SPAN><SPAN class=mn>0</SPAN></FONT></SPAN></TD></TR></TBODY></TABLE><SPAN id=id7714450L> <TABLE class=lr> <TBODY> <TR> <TD><FONT size=2><FONT face=symbol>æ</FONT><BR><FONT face=symbol>ç</FONT><BR><FONT face=symbol>è</FONT></FONT></TD></TR></TBODY></TABLE></SPAN><SPAN id=id7714450M><SPAN class=mrow id=id7714453><EM><FONT size=2><SPAN class=mrow id=id7714454><SPAN><SPAN class=mi1>b</SPAN></SPAN><SUB><SPAN class=mi1>i</SPAN></SUB></SPAN><SPAN class=mo id=id7714466></SPAN></FONT></EM><SPAN class=mrow id=id7714469><FONT size=2> </FONT> <TABLE class=mfrac style="FONT-SIZE: 75%"> <TBODY> <TR class=mfraca id=aid7714470> <TD class=mfracaa><SPAN class=mrow id=id7714471><FONT size=2><SPAN><SPAN class=mo id=id7714473>d</SPAN></SPAN><SUP><SPAN class=mrow id=id7714476><SPAN class=mi1><EM>i</EM></SPAN></SPAN></SUP></FONT></SPAN></TD></TR> <TR class=mfracb id=bid7714470 style="HEIGHT: 22px"> <TD><SPAN class=mrow id=id7714480><FONT size=2><SPAN class=mo id=id7714481>d</SPAN><EM><SPAN><SPAN class=mrow id=id7714485><SPAN class=mi1>t</SPAN></SPAN></SPAN><SUP><SPAN class=mrow id=id7714490><SPAN class=mi1>i</SPAN></SPAN></SUP></EM></FONT></SPAN></TD></TR></TBODY></TABLE> <SCRIPT type=text/javascript> if ( aid7714470.offsetHeight > bid7714470.offsetHeight ) bid7714470.style.setExpression("height",aid7714470.offsetHeight ); else aid7714470.style.setExpression("height",bid7714470.offsetHeight ); </SCRIPT> <SPAN class=mrow id=id7714493><SPAN class=mrow id=id7714496><FONT size=2><EM><SPAN class=mi1>f</SPAN><SPAN class=mo id=id7714499></SPAN></EM><SPAN id=id7714501L>(</SPAN><SPAN id=id7714501M><SPAN class=mi1><EM>t</EM></SPAN></SPAN><SPAN id=id7714501R>)</SPAN> <SCRIPT type=text/javascript> var mrowH = id7714501M.offsetHeight; mrowStretch(id7714501L,'æ','ç','ç','è'); mrowStretch(id7714501R,'ö','÷','÷','ø');</SCRIPT> </FONT></SPAN></SPAN></SPAN></SPAN></SPAN><SPAN id=id7714450R> <TABLE class=lr> <TBODY> <TR> <TD><FONT size=2><FONT face=symbol>ö</FONT><BR><FONT face=symbol>÷</FONT><BR><FONT face=symbol>ø</FONT></FONT></TD></TR></TBODY></TABLE></SPAN> <SCRIPT type=text/javascript> var mrowH = id7714450M.offsetHeight; mrowStretch(id7714450L,'æ','ç','ç','è'); mrowStretch(id7714450R,'ö','÷','÷','ø');</SCRIPT> </SPAN></SPAN></SPAN></SPAN><FONT size=2><SPAN class=equation-number>(1)</SPAN> </FONT></DIV><FONT size=2>con <SPAN class=mrow id=id7713723><SPAN class=mrow id=id7713717><SPAN class=mrow id=id7714695><SPAN class=mrow id=id7714696><EM><SPAN><SPAN class=mi1>a</SPAN></SPAN><SUB><SPAN class=mi1>n</SPAN></SUB></EM></SPAN><SPAN class=mo id=id7714707>=</SPAN><SPAN class=mn>1</SPAN></SPAN></SPAN></SPAN>. </FONT></DIV> <DIV class=para id=p3><FONT size=2>Es fácil mostra que estas ecuaciones definen un sistema que es lineal e invariante en el tiempo. Entonces una pregunta natural es ¿cómo encontrar la respuesta al impulso del sistema <SPAN class=mrow id=id7714692><SPAN class=mrow id=id7714752><SPAN class=mrow id=id7714756><EM><SPAN class=mi1>y</SPAN><SPAN class=mo id=id7714759></SPAN></EM><SPAN id=id7714761L>(</SPAN><SPAN id=id7714761M><SPAN class=mi1><EM>t</EM></SPAN></SPAN><SPAN id=id7714761R>)</SPAN> <SCRIPT type=text/javascript> var mrowH = id7714761M.offsetHeight; mrowStretch(id7714761L,'æ','ç','ç','è'); mrowStretch(id7714761R,'ö','÷','÷','ø');</SCRIPT> </SPAN></SPAN></SPAN>para una entrada <SPAN class=mrow id=id7714783><SPAN class=mrow id=id7714786><SPAN class=mrow id=id7714792><EM><SPAN class=mi1>f</SPAN><SPAN class=mo id=id7714796></SPAN></EM><SPAN id=id7714798L>(</SPAN><SPAN id=id7714798M><SPAN class=mi1><EM>t</EM></SPAN></SPAN><SPAN id=id7714798R>)</SPAN> <SCRIPT type=text/javascript> var mrowH = id7714798M.offsetHeight; mrowStretch(id7714798L,'æ','ç','ç','è'); mrowStretch(id7714798R,'ö','÷','÷','ø');</SCRIPT> </SPAN></SPAN></SPAN>?. Recordando que tal solución se puede escribir como: <SPAN class=mrow id=id7714820><SPAN class=mrow id=id7714822><SPAN class=mrow id=id7714836><SPAN class=mrow id=id7714837><EM><SPAN class=mi1>y</SPAN><SPAN class=mo id=id7714840></SPAN></EM><SPAN id=id7714842L>(</SPAN><SPAN id=id7714842M><SPAN class=mi1><EM>t</EM></SPAN></SPAN><SPAN id=id7714842R>)</SPAN> <SCRIPT type=text/javascript> var mrowH = id7714842M.offsetHeight; mrowStretch(id7714842L,'æ','ç','ç','è'); mrowStretch(id7714842R,'ö','÷','÷','ø');</SCRIPT> </SPAN><SPAN class=mo id=id7714845>=</SPAN><SPAN class=mrow id=id7714848><SPAN class=mrow id=id7714850><EM><SPAN class=mrow id=id7714851><SPAN><SPAN class=mi1>y</SPAN></SPAN><SUB><SPAN class=mi1>i</SPAN></SUB></SPAN><SPAN class=mo id=id7714862></SPAN></EM><SPAN id=id7714864L>(</SPAN><SPAN id=id7714864M><SPAN class=mi1><EM>t</EM></SPAN></SPAN><SPAN id=id7714864R>)</SPAN> <SCRIPT type=text/javascript> var mrowH = id7714864M.offsetHeight; mrowStretch(id7714864L,'æ','ç','ç','è'); mrowStretch(id7714864R,'ö','÷','÷','ø');</SCRIPT> </SPAN><SPAN class=mo id=id7714868>+</SPAN><SPAN class=mrow id=id7714873><EM><SPAN class=mrow id=id7714874><SPAN><SPAN class=mi1>y</SPAN></SPAN><SUB><SPAN class=mi1>s</SPAN></SUB></SPAN><SPAN class=mo id=id7714885></SPAN></EM><SPAN id=id7714888L>(</SPAN><SPAN id=id7714888M><SPAN class=mi1><EM>t</EM></SPAN></SPAN><SPAN id=id7714888R>)</SPAN> <SCRIPT type=text/javascript> var mrowH = id7714888M.offsetHeight; mrowStretch(id7714888L,'æ','ç','ç','è'); mrowStretch(id7714888R,'ö','÷','÷','ø');</SCRIPT> </SPAN></SPAN></SPAN></SPAN></SPAN>Nos referimos a <SPAN class=mrow id=id7714974><SPAN class=mrow id=id7714871><SPAN class=mrow id=id7714978><EM><SPAN class=mrow id=id7714980><SPAN><SPAN class=mi1>y</SPAN></SPAN><SUB><SPAN class=mi1>i</SPAN></SUB></SPAN><SPAN class=mo id=id7714990></SPAN></EM><SPAN id=id7714992L>(</SPAN><SPAN id=id7714992M><SPAN class=mi1><EM>t</EM></SPAN></SPAN><SPAN id=id7714992R>)</SPAN> <SCRIPT type=text/javascript> var mrowH = id7714992M.offsetHeight; mrowStretch(id7714992L,'æ','ç','ç','è'); mrowStretch(id7714992R,'ö','÷','÷','ø');</SCRIPT> </SPAN></SPAN></SPAN>como la <SPAN class=term><STRONG>respuesta de salida-cero</STRONG></SPAN> -- la solución homogenea debido a las condiciones iniciales del sistema. Nos referimos a <SPAN class=mrow id=id7715034><SPAN class=mrow id=id7715036><SPAN class=mrow id=id7715044><EM><SPAN class=mrow id=id7715045><SPAN><SPAN class=mi1>y</SPAN></SPAN><SUB><SPAN class=mi1>s</SPAN></SUB></SPAN><SPAN class=mo id=id7715056></SPAN></EM><SPAN id=id7715059L>(</SPAN><SPAN id=id7715059M><SPAN class=mi1><EM>t</EM></SPAN></SPAN><SPAN id=id7715059R>)</SPAN> <SCRIPT type=text/javascript> var mrowH = id7715059M.offsetHeight; mrowStretch(id7715059L,'æ','ç','ç','è'); mrowStretch(id7715059R,'ö','÷','÷','ø');</SCRIPT> </SPAN></SPAN></SPAN>como la <SPAN class=term><STRONG>respuesta de estado-cero</STRONG></SPAN> -- la solución particular en respuesta a la entrada <SPAN class=mrow id=id7715099><SPAN class=mrow id=id7715101><SPAN class=mrow id=id7715109><EM><SPAN class=mi1>f</SPAN><SPAN class=mo id=id7715112></SPAN></EM><SPAN id=id7715115L>(</SPAN><SPAN id=id7715115M><SPAN class=mi1><EM>t</EM></SPAN></SPAN><SPAN id=id7715115R>)</SPAN> <SCRIPT type=text/javascript> var mrowH = id7715115M.offsetHeight; mrowStretch(id7715115L,'æ','ç','ç','è'); mrowStretch(id7715115R,'ö','÷','÷','ø');</SCRIPT> </SPAN></SPAN></SPAN>. Ahora veremos como resolver cada una de estos componentes de la respuesta del sistema. </FONT></DIV> <DIV class=section id=sec2> <H2><FONT size=2>Encontrando la Respuesta de Entrada-Cero</FONT></H2> <DIV class=para id=p1_sec2><FONT size=2>La respuesta de la entrada-cero, <SPAN class=mrow id=id7715152><SPAN class=mrow id=id7715155><SPAN class=mrow id=id7715164><EM><SPAN class=mrow id=id7715166><SPAN><SPAN class=mi1>y</SPAN></SPAN><SUB><SPAN class=mi1>i</SPAN></SUB></SPAN><SPAN class=mo id=id7715177></SPAN></EM><SPAN id=id7715179L>(</SPAN><SPAN id=id7715179M><SPAN class=mi1><EM>t</EM></SPAN></SPAN><SPAN id=id7715179R>)</SPAN> <SCRIPT type=text/javascript> var mrowH = id7715179M.offsetHeight; mrowStretch(id7715179L,'æ','ç','ç','è'); mrowStretch(id7715179R,'ö','÷','÷','ø');</SCRIPT> </SPAN></SPAN></SPAN>, es la respuesta del sistema debido solo a las condiciones iniciales . </FONT></DIV> <DIV class=example id=eg1><FONT size=2><STRONG><SPAN class=example-before><FONT color=#336699>Example 1: </FONT></SPAN><SPAN class=example-name>Respuesta de la Entrada-Cero </SPAN></STRONG> </FONT> <DIV class=para id=p1_eg1><FONT size=2>Poner un voltaje a través de una capacitor en un circuito dibujado a continuación y deje todo lo demás solo. </FONT></DIV> <TABLE class=figure id=fig2 cellSpacing=0 cellPadding=0 width="50%" align=center border=0> <TBODY> <TR> <TD> <DIV class=inner-figure><FONT size=2><IMG class=media src= "http://cnx.org/content/m12985/latest/image1.png" alt="image" />" ></FONT></DIV> <DIV class=caption><SPAN class=caption-before><STRONG><FONT size=2>Figure 2: </FONT></STRONG></SPAN></DIV></TD></TR></TBODY></TABLE></DIV> <DIV class=example id=eg2><FONT size=2><STRONG><SPAN class=example-before><FONT color=#336699>Example 2: </FONT></SPAN><SPAN class=example-name>Respuesta de la Entrada-Cero </SPAN></STRONG> </FONT> <DIV class=para id=p1_eg2><FONT size=2>Imagine una masa unidia a un resorte como se muestra a continuación. Cuando jala la masa hacia abajo y la suelte, usted tiene un ejemplo de la respuesta de la entrada-cero. </FONT></DIV> <TABLE class=figure id=fig3 cellSpacing=0 cellPadding=0 width="50%" align=center border=0> <TBODY> <TR> <TD> <DIV class=inner-figure><FONT size=2><IMG class=media src= "http://cnx.org/content/m12985/latest/image1.png" alt="image" />" ></FONT></DIV> <DIV class=caption><SPAN class=caption-before><STRONG><FONT size=2>Figure 3: </FONT></STRONG></SPAN></DIV></TD></TR></TBODY></TABLE></DIV> <DIV class=para id=p4><FONT size=2>No hay entrada, así que resolvemos para: <SPAN class=mrow id=id7715375><SPAN class=mrow id=id7715378><SPAN class=mrow id=id7715386><SPAN class=mrow id=id7715388><SPAN><SPAN class=mi1><EM>y</EM></SPAN></SPAN><SUB><SPAN class=mn>0</SPAN></SUB></SPAN><SPAN class=mo id=id7715399></SPAN><SPAN id=id7715401L>(</SPAN><SPAN id=id7715401M><SPAN class=mi1><EM>t</EM></SPAN></SPAN><SPAN id=id7715401R>)</SPAN> <SCRIPT type=text/javascript> var mrowH = id7715401M.offsetHeight; mrowStretch(id7715401L,'æ','ç','ç','è'); mrowStretch(id7715401R,'ö','÷','÷','ø');</SCRIPT> </SPAN></SPAN></SPAN>tal que </FONT> <DIV class=equation id=eq2><SPAN class=mrow id=id7712563><SPAN class=mrow id=id7712569><SPAN class=mrow id=id7712578><SPAN class=mrow id=id7712587><SPAN class=mrow id=id7712588> <TABLE class=munderover> <TBODY> <TR> <TD><SPAN class=mrow id=id7712604><SPAN class=mi1><EM><FONT size=2>n</FONT></EM></SPAN></SPAN></TD></TR> <TR> <TD><SPAN class=mo id=id7712591><FONT size=2>∑</FONT></SPAN></TD></TR> <TR> <TD><SPAN class=mrow id=id7712593><FONT size=2><SPAN class=mi1><EM>i</EM></SPAN><SPAN class=mo id=id7712599>=</SPAN><SPAN class=mn>0</SPAN></FONT></SPAN></TD></TR></TBODY></TABLE><SPAN id=id7712609L> <TABLE class=lr> <TBODY> <TR> <TD><FONT size=2><FONT face=symbol>æ</FONT><BR><FONT face=symbol>ç</FONT><BR><FONT face=symbol>è</FONT></FONT></TD></TR></TBODY></TABLE></SPAN><SPAN id=id7712609M><SPAN class=mrow id=id7712614><EM><FONT size=2><SPAN class=mrow id=id7712615><SPAN><SPAN class=mi1>a</SPAN></SPAN><SUB><SPAN class=mi1>i</SPAN></SUB></SPAN><SPAN class=mo id=id7712626></SPAN></FONT></EM><SPAN class=mrow id=id7712634><FONT size=2> </FONT> <TABLE class=mfrac style="FONT-SIZE: 75%"> <TBODY> <TR class=mfraca id=aid7712635> <TD class=mfracaa><SPAN class=mrow id=id7712636><FONT size=2><SPAN><SPAN class=mo id=id7712638>d</SPAN></SPAN><SUP><SPAN class=mrow id=id7712640><SPAN class=mi1><EM>i</EM></SPAN></SPAN></SUP></FONT></SPAN></TD></TR> <TR class=mfracb id=bid7712635 style="HEIGHT: 22px"> <TD><SPAN class=mrow id=id7712646><FONT size=2><SPAN class=mo id=id7712647>d</SPAN><EM><SPAN><SPAN class=mrow id=id7712650><SPAN class=mi1>t</SPAN></SPAN></SPAN><SUP><SPAN class=mrow id=id7712656><SPAN class=mi1>i</SPAN></SPAN></SUP></EM></FONT></SPAN></TD></TR></TBODY></TABLE> <SCRIPT type=text/javascript> if ( aid7712635.offsetHeight > bid7712635.offsetHeight ) bid7712635.style.setExpression("height",aid7712635.offsetHeight ); else aid7712635.style.setExpression("height",bid7712635.offsetHeight ); </SCRIPT> <SPAN class=mrow id=id7712659><SPAN class=mrow id=id7712665><FONT size=2><SPAN class=mrow id=id7712666><SPAN><SPAN class=mi1><EM>y</EM></SPAN></SPAN><SUB><SPAN class=mn>0</SPAN></SUB></SPAN><SPAN class=mo id=id7712677></SPAN><SPAN id=id7712680L>(</SPAN><SPAN id=id7712680M><SPAN class=mi1><EM>t</EM></SPAN></SPAN><SPAN id=id7712680R>)</SPAN> <SCRIPT type=text/javascript> var mrowH = id7712680M.offsetHeight; mrowStretch(id7712680L,'æ','ç','ç','è'); mrowStretch(id7712680R,'ö','÷','÷','ø');</SCRIPT> </FONT></SPAN></SPAN></SPAN></SPAN></SPAN><SPAN id=id7712609R> <TABLE class=lr> <TBODY> <TR> <TD><FONT size=2><FONT face=symbol>ö</FONT><BR><FONT face=symbol>÷</FONT><BR><FONT face=symbol>ø</FONT></FONT></TD></TR></TBODY></TABLE></SPAN> <SCRIPT type=text/javascript> var mrowH = id7712609M.offsetHeight; mrowStretch(id7712609L,'æ','ç','ç','è'); mrowStretch(id7712609R,'ö','÷','÷','ø');</SCRIPT> </SPAN><FONT size=2><SPAN class=mo id=id7712683>=</SPAN><SPAN class=mn>0</SPAN></FONT></SPAN><FONT size=2><SPAN class=mo id=id7712687>  </SPAN><SPAN class=mo id=id7712689>,</SPAN><SPAN class=mo id=id7712691>  </SPAN><SPAN class=mrow id=id7712698><SPAN class=mrow id=id7712699><EM><SPAN><SPAN class=mi1>a</SPAN></SPAN><SUB><SPAN class=mi1>n</SPAN></SUB></EM></SPAN><SPAN class=mo id=id7712709>=</SPAN><SPAN class=mn>1</SPAN></SPAN><SPAN class=mo id=id7712716>  </SPAN></FONT></SPAN></SPAN></SPAN><FONT size=2> <SPAN class=equation-number>(2)</SPAN> </FONT></DIV><FONT size=2>Si <SPAN class=mrow id=id7712573><SPAN class=mrow id=id7712661><SPAN class=mi1><EM>D</EM></SPAN></SPAN></SPAN> es el operador derivado, podemos escribir la ecuación anterior como: </FONT> <DIV class=equation id=eq3><FONT size=2><SPAN class=mrow id=id7712896><SPAN class=mrow id=id7712900><SPAN class=mrow id=id7712915><SPAN class=mrow id=id7715814><SPAN id=id7715815L>(</SPAN><SPAN id=id7715815M><SPAN class=mrow id=id7715819><EM><SPAN><SPAN class=mi1>D</SPAN></SPAN><SUP><SPAN class=mi1>n</SPAN></SUP></EM><SPAN class=mo id=id7715830>+</SPAN><SPAN class=mrow id=id7715833><SPAN class=mrow id=id7715834><SPAN><SPAN class=mi1><EM>a</EM></SPAN></SPAN><SUB><SPAN class=mrow id=id7715841><SPAN class=mi1><EM>n</EM></SPAN><SPAN class=mo id=id7715848>−</SPAN><SPAN class=mn>1</SPAN></SPAN></SUB></SPAN><SPAN class=mo id=id7715857></SPAN><SPAN><SPAN class=mi1><EM>D</EM></SPAN></SPAN><SUP><SPAN class=mrow id=id7715873><SPAN class=mi1><EM>n</EM></SPAN><SPAN class=mo id=id7715877>−</SPAN><SPAN class=mn>1</SPAN></SPAN></SUP></SPAN><SPAN class=mo id=id7715882>+</SPAN><SPAN class=mi1><EM>…</EM></SPAN><SPAN class=mo id=id7715887>+</SPAN><SPAN class=mrow id=id7715889><SPAN><SPAN class=mi1><EM>a</EM></SPAN></SPAN><SUB><SPAN class=mn>0</SPAN></SUB></SPAN></SPAN></SPAN><SPAN id=id7715815R>)</SPAN> <SCRIPT type=text/javascript> var mrowH = id7715815M.offsetHeight; mrowStretch(id7715815L,'æ','ç','ç','è'); mrowStretch(id7715815R,'ö','÷','÷','ø');</SCRIPT> <SPAN class=mo id=id7715901></SPAN><SPAN class=mrow id=id7715903><SPAN class=mrow id=id7715904><SPAN><SPAN class=mi1><EM>y</EM></SPAN></SPAN><SUB><SPAN class=mn>0</SPAN></SUB></SPAN><SPAN class=mo id=id7715915></SPAN><SPAN id=id7715918L>(</SPAN><SPAN id=id7715918M><SPAN class=mi1><EM>t</EM></SPAN></SPAN><SPAN id=id7715918R>)</SPAN> <SCRIPT type=text/javascript> var mrowH = id7715918M.offsetHeight; mrowStretch(id7715918L,'æ','ç','ç','è'); mrowStretch(id7715918R,'ö','÷','÷','ø');</SCRIPT> </SPAN></SPAN></SPAN><SPAN class=mrow id=id7715921><SPAN class=mo id=id7715922>=</SPAN></SPAN><SPAN class=mrow id=id7715924><SPAN class=mn>0</SPAN></SPAN></SPAN></SPAN> <SPAN class=equation-number>(3)</SPAN> </FONT></DIV><FONT size=2>Puesto que necesitamos una suma de varios <SPAN class=mrow id=id7712914><SPAN class=mrow id=id7715862><SPAN class=mrow id=id7715860><SPAN class=mrow id=id7715861><SPAN><SPAN class=mi1><EM>y</EM></SPAN></SPAN><SUB><SPAN class=mn>0</SPAN></SUB></SPAN><SPAN class=mo id=id7716081></SPAN><SPAN id=id7716084L>(</SPAN><SPAN id=id7716084M><SPAN class=mi1><EM>t</EM></SPAN></SPAN><SPAN id=id7716084R>)</SPAN> <SCRIPT type=text/javascript> var mrowH = id7716084M.offsetHeight; mrowStretch(id7716084L,'æ','ç','ç','è'); mrowStretch(id7716084R,'ö','÷','÷','ø');</SCRIPT> </SPAN></SPAN></SPAN>'s dervados para ser <SPAN class=mrow id=id7716117><SPAN class=mrow id=id7716121><SPAN class=mn>0</SPAN></SPAN></SPAN> <I>para todo</I> <SPAN class=mrow id=id7716135><SPAN class=mrow id=id7716138><SPAN class=mi1><EM>t</EM></SPAN></SPAN></SPAN>, entonces </FONT><SPAN class=mrow id=id7716150><SPAN class=mrow id=id7716154><SPAN style="COLOR: red"><FONT size=2><apply></FONT></SPAN><SPAN class=mrow id=id7716165><SPAN id=id7716167L><FONT size=2> </FONT> <TABLE class=lr> <TBODY> <TR> <TD><FONT size=2><FONT face=symbol>ì</FONT><BR><FONT face=symbol>ï</FONT><BR><FONT face=symbol>í</FONT><BR><FONT face=symbol>ï</FONT><BR><FONT face=symbol>î</FONT></FONT></TD></TR></TBODY></TABLE></SPAN><SPAN id=id7716167M><FONT size=2><SPAN class=mrow id=id7716180><SPAN class=mrow id=id7716182><SPAN><SPAN class=mi1><EM>y</EM></SPAN></SPAN><SUB><SPAN class=mn>0</SPAN></SUB></SPAN><SPAN class=mo id=id7716193></SPAN><SPAN id=id7716195L>(</SPAN><SPAN id=id7716195M><SPAN class=mi1><EM>t</EM></SPAN></SPAN><SPAN id=id7716195R>)</SPAN> <SCRIPT type=text/javascript> var mrowH = id7716195M.offsetHeight; mrowStretch(id7716195L,'æ','ç','ç','è'); mrowStretch(id7716195R,'ö','÷','÷','ø');</SCRIPT> </SPAN><SPAN id=id7716180X1>,</SPAN></FONT><SPAN class=mrow id=id7716200><FONT size=2> </FONT> <TABLE class=mfrac style="FONT-SIZE: 75%"> <TBODY> <TR class=mfraca id=aid7716201> <TD class=mfracaa><SPAN class=mo id=id7716202><FONT size=2>d</FONT></SPAN></TD></TR> <TR class=mfracb id=bid7716201 style="HEIGHT: 20px"> <TD><SPAN class=mrow id=id7716204><FONT size=2><SPAN class=mo id=id7716205>d</SPAN><SPAN class=mi1><EM>t</EM></SPAN></FONT></SPAN></TD></TR></TBODY></TABLE> <SCRIPT type=text/javascript> if ( aid7716201.offsetHeight > bid7716201.offsetHeight ) bid7716201.style.setExpression("height",aid7716201.offsetHeight ); else aid7716201.style.setExpression("height",bid7716201.offsetHeight ); </SCRIPT> <SPAN class=mrow id=id7716212><SPAN class=mrow id=id7716218><FONT size=2><SPAN class=mrow id=id7716219><SPAN><SPAN class=mi1><EM>y</EM></SPAN></SPAN><SUB><SPAN class=mn>0</SPAN></SUB></SPAN><SPAN class=mo id=id7716230></SPAN><SPAN id=id7716232L>(</SPAN><SPAN id=id7716232M><SPAN class=mi1><EM>t</EM></SPAN></SPAN><SPAN id=id7716232R>)</SPAN> <SCRIPT type=text/javascript> var mrowH = id7716232M.offsetHeight; mrowStretch(id7716232L,'æ','ç','ç','è'); mrowStretch(id7716232R,'ö','÷','÷','ø');</SCRIPT> </FONT></SPAN></SPAN></SPAN><SPAN id=id7716200X2><FONT size=2>,</FONT></SPAN><SPAN class=mrow id=id7716236><FONT size=2> </FONT> <TABLE class=mfrac style="FONT-SIZE: 75%"> <TBODY> <TR class=mfraca id=aid7716237> <TD class=mfracaa><SPAN class=mrow id=id7716238><FONT size=2><SPAN><SPAN class=mo id=id7716240>d</SPAN></SPAN><SUP><SPAN class=mrow id=id7716242><SPAN class=mn>2</SPAN></SPAN></SUP></FONT></SPAN></TD></TR> <TR class=mfracb id=bid7716237 style="HEIGHT: 22px"> <TD><SPAN class=mrow id=id7716247><FONT size=2><SPAN class=mo id=id7716248>d</SPAN><SPAN><SPAN class=mrow id=id7716252><SPAN class=mi1><EM>t</EM></SPAN></SPAN></SPAN><SUP><SPAN class=mrow id=id7716257><SPAN class=mn>2</SPAN></SPAN></SUP></FONT></SPAN></TD></TR></TBODY></TABLE> <SCRIPT type=text/javascript> if ( aid7716237.offsetHeight > bid7716237.offsetHeight ) bid7716237.style.setExpression("height",aid7716237.offsetHeight ); else aid7716237.style.setExpression("height",bid7716237.offsetHeight ); </SCRIPT> <SPAN class=mrow id=id7716260><SPAN class=mrow id=id7716264><FONT size=2><SPAN class=mrow id=id7716265><SPAN><SPAN class=mi1><EM>y</EM></SPAN></SPAN><SUB><SPAN class=mn>0</SPAN></SUB></SPAN><SPAN class=mo id=id7716276></SPAN><SPAN id=id7716278L>(</SPAN><SPAN id=id7716278M><SPAN class=mi1><EM>t</EM></SPAN></SPAN><SPAN id=id7716278R>)</SPAN> <SCRIPT type=text/javascript> var mrowH = id7716278M.offsetHeight; mrowStretch(id7716278L,'æ','ç','ç','è'); mrowStretch(id7716278R,'ö','÷','÷','ø');</SCRIPT> </FONT></SPAN></SPAN></SPAN><FONT size=2><SPAN id=id7716236X3>,</SPAN><SPAN class=mi1><EM>…</EM></SPAN></FONT></SPAN><SPAN id=id7716167R><FONT size=2> </FONT> <TABLE class=lr> <TBODY> <TR> <TD><FONT size=2><FONT face=symbol>ü</FONT><BR><FONT face=symbol>ú</FONT><BR><FONT face=symbol>ý</FONT><BR><FONT face=symbol>ú</FONT><BR><FONT face=symbol>þ</FONT></FONT></TD></TR></TBODY></TABLE></SPAN> <SCRIPT type=text/javascript> var mrowH = id7716167M.offsetHeight; mrowStretch(id7716167L,'ì','ï','í','î'); mrowStretch(id7716167R,'ü','ú','ý','þ');</SCRIPT> </SPAN><SPAN style="COLOR: red"><FONT size=2></apply></FONT></SPAN></SPAN></SPAN><FONT size=2> deben de ser todos de la misma forma. </FONT></DIV> <DIV class=para id=p2_sec2><FONT size=2>Solo el exponencial, <SPAN class=mrow id=id7716214><SPAN class=mrow id=id7716176><EM><SPAN><SPAN class=mi1>ⅇ</SPAN></SPAN><SUP><SPAN class=mrow id=id7716433><SPAN class=mi1>s</SPAN><SPAN class=mo id=id7716436></SPAN><SPAN class=mi1>t</SPAN></SPAN></SUP></EM></SPAN></SPAN> donde <SPAN class=mrow id=id7716465><SPAN class=mrow id=id7716466><SPAN class=mrow id=id7716473><SPAN class=mi1><EM>s</EM></SPAN><SPAN class=mo id=id7716477>∈</SPAN><SPAN class=mi>&#x2102;</SPAN></SPAN></SPAN></SPAN>, tiene esta propiedad (véase un libro de texto de Ecuaciones Diferenciales para más detalles). Así que asumimos que, </FONT> <DIV class=equation id=eq4><FONT size=2><SPAN class=mrow id=id7716509><SPAN class=mrow id=id7716512><SPAN class=mrow id=id7716521><SPAN class=mrow id=id7716531><SPAN class=mrow id=id7716532><SPAN class=mrow id=id7716533><SPAN><SPAN class=mi1><EM>y</EM></SPAN></SPAN><SUB><SPAN class=mn>0</SPAN></SUB></SPAN><SPAN class=mo id=id7716544></SPAN><SPAN id=id7716546L>(</SPAN><SPAN id=id7716546M><SPAN class=mi1><EM>t</EM></SPAN></SPAN><SPAN id=id7716546R>)</SPAN> <SCRIPT type=text/javascript> var mrowH = id7716546M.offsetHeight; mrowStretch(id7716546L,'æ','ç','ç','è'); mrowStretch(id7716546R,'ö','÷','÷','ø');</SCRIPT> </SPAN><SPAN class=mo id=id7716549>=</SPAN><SPAN class=mrow id=id7716553><EM><SPAN class=mi1>c</SPAN><SPAN class=mo id=id7716556></SPAN><SPAN><SPAN class=mi1>ⅇ</SPAN></SPAN><SUP><SPAN class=mrow id=id7716567><SPAN class=mi1>s</SPAN><SPAN class=mo id=id7716570></SPAN><SPAN class=mi1>t</SPAN></SPAN></SUP></EM></SPAN></SPAN><SPAN class=mo id=id7716574>  </SPAN><SPAN class=mo id=id7716576>,</SPAN><SPAN class=mo id=id7716578>  </SPAN><SPAN class=mrow id=id7716584><SPAN class=mi1><EM>c</EM></SPAN><SPAN class=mo id=id7716588>≠</SPAN><SPAN class=mn>0</SPAN></SPAN><SPAN class=mo id=id7716594>  </SPAN></SPAN></SPAN></SPAN> <SPAN class=equation-number>(4)</SPAN> </FONT></DIV><FONT size=2>para algun <SPAN class=mrow id=id7716529><SPAN class=mrow id=id7716558><SPAN class=mi1><EM>c</EM></SPAN></SPAN></SPAN> y <SPAN class=mrow id=id7716709><SPAN class=mrow id=id7716713><SPAN class=mi1><EM>s</EM></SPAN></SPAN></SPAN>. </FONT></DIV> <DIV class=para id=p3_sec2><FONT size=2>Since </FONT><SPAN class=mrow id=id7716733><SPAN class=mrow id=id7716737><SPAN class=mrow id=id7716752><SPAN class=mrow id=id7716753> <TABLE class=mfrac style="FONT-SIZE: 75%"> <TBODY> <TR class=mfraca id=aid7716755> <TD class=mfracaa><SPAN class=mo id=id7716756><FONT size=2>d</FONT></SPAN></TD></TR> <TR class=mfracb id=bid7716755 style="HEIGHT: 20px"> <TD><SPAN class=mrow id=id7716758><FONT size=2><SPAN class=mo id=id7716759>d</SPAN><SPAN class=mi1><EM>t</EM></SPAN></FONT></SPAN></TD></TR></TBODY></TABLE> <SCRIPT type=text/javascript> if ( aid7716755.offsetHeight > bid7716755.offsetHeight ) bid7716755.style.setExpression("height",aid7716755.offsetHeight ); else aid7716755.style.setExpression("height",bid7716755.offsetHeight ); </SCRIPT> <SPAN class=mrow id=id7716766><SPAN class=mrow id=id7716771><FONT size=2><SPAN class=mrow id=id7716772><SPAN><SPAN class=mi1><EM>y</EM></SPAN></SPAN><SUB><SPAN class=mn>0</SPAN></SUB></SPAN><SPAN class=mo id=id7716784></SPAN><SPAN id=id7716786L>(</SPAN><SPAN id=id7716786M><SPAN class=mi1><EM>t</EM></SPAN></SPAN><SPAN id=id7716786R>)</SPAN> <SCRIPT type=text/javascript> var mrowH = id7716786M.offsetHeight; mrowStretch(id7716786L,'æ','ç','ç','è'); mrowStretch(id7716786R,'ö','÷','÷','ø');</SCRIPT> </FONT></SPAN></SPAN></SPAN><FONT size=2><SPAN class=mo id=id7716789>=</SPAN><SPAN class=mrow id=id7716792><EM><SPAN class=mi1>c</SPAN><SPAN class=mo id=id7716795></SPAN><SPAN class=mi1>s</SPAN><SPAN class=mo id=id7716799></SPAN><SPAN><SPAN class=mi1>ⅇ</SPAN></SPAN><SUP><SPAN class=mrow id=id7716808><SPAN class=mi1>s</SPAN><SPAN class=mo id=id7716811></SPAN><SPAN class=mi1>t</SPAN></SPAN></SUP></EM></SPAN></FONT></SPAN></SPAN></SPAN><FONT size=2>, </FONT><SPAN class=mrow id=id7716900><SPAN class=mrow id=id7716750><SPAN class=mrow id=id7716907><SPAN class=mrow id=id7716909> <TABLE class=mfrac style="FONT-SIZE: 75%"> <TBODY> <TR class=mfraca id=aid7716911> <TD class=mfracaa><SPAN class=mrow id=id7716912><FONT size=2><SPAN><SPAN class=mo id=id7716914>d</SPAN></SPAN><SUP><SPAN class=mrow id=id7716916><SPAN class=mn>2</SPAN></SPAN></SUP></FONT></SPAN></TD></TR> <TR class=mfracb id=bid7716911 style="HEIGHT: 22px"> <TD><SPAN class=mrow id=id7716922><FONT size=2><SPAN class=mo id=id7716923>d</SPAN><SPAN><SPAN class=mrow id=id7716927><SPAN class=mi1><EM>t</EM></SPAN></SPAN></SPAN><SUP><SPAN class=mrow id=id7716932><SPAN class=mn>2</SPAN></SPAN></SUP></FONT></SPAN></TD></TR></TBODY></TABLE> <SCRIPT type=text/javascript> if ( aid7716911.offsetHeight > bid7716911.offsetHeight ) bid7716911.style.setExpression("height",aid7716911.offsetHeight ); else aid7716911.style.setExpression("height",bid7716911.offsetHeight ); </SCRIPT> <SPAN class=mrow id=id7716936><SPAN class=mrow id=id7716941><FONT size=2><SPAN class=mrow id=id7716942><SPAN><SPAN class=mi1><EM>y</EM></SPAN></SPAN><SUB><SPAN class=mn>0</SPAN></SUB></SPAN><SPAN class=mo id=id7716954></SPAN><SPAN id=id7716956L>(</SPAN><SPAN id=id7716956M><SPAN class=mi1><EM>t</EM></SPAN></SPAN><SPAN id=id7716956R>)</SPAN> <SCRIPT type=text/javascript> var mrowH = id7716956M.offsetHeight; mrowStretch(id7716956L,'æ','ç','ç','è'); mrowStretch(id7716956R,'ö','÷','÷','ø');</SCRIPT> </FONT></SPAN></SPAN></SPAN><FONT size=2><SPAN class=mo id=id7716959>=</SPAN><SPAN class=mrow id=id7716962><EM><SPAN class=mi1>c</SPAN><SPAN class=mo id=id7716965></SPAN><SPAN><SPAN class=mi1>s</SPAN></SPAN></EM><SUP><SPAN class=mn>2</SPAN></SUP><SPAN class=mo id=id7716974></SPAN><EM><SPAN><SPAN class=mi1>ⅇ</SPAN></SPAN><SUP><SPAN class=mrow id=id7716981><SPAN class=mi1>s</SPAN><SPAN class=mo id=id7716984></SPAN><SPAN class=mi1>t</SPAN></SPAN></SUP></EM></SPAN></FONT></SPAN></SPAN></SPAN><FONT size=2>, … tenemos <SPAN class=mrow id=id7717092><SPAN class=mrow id=id7716940><SPAN class=mrow id=id7717100><SPAN class=mrow id=id7717101><SPAN id=id7717102L>(</SPAN><SPAN id=id7717102M><SPAN class=mrow id=id7717110><EM><SPAN><SPAN class=mi1>D</SPAN></SPAN><SUP><SPAN class=mi1>n</SPAN></SUP></EM><SPAN class=mo id=id7717122>+</SPAN><SPAN class=mrow id=id7717125><SPAN class=mrow id=id7717126><SPAN><SPAN class=mi1><EM>a</EM></SPAN></SPAN><SUB><SPAN class=mrow id=id7717134><SPAN class=mi1><EM>n</EM></SPAN><SPAN class=mo id=id7717140>−</SPAN><SPAN class=mn>1</SPAN></SPAN></SUB></SPAN><SPAN class=mo id=id7717150></SPAN><SPAN><SPAN class=mi1><EM>D</EM></SPAN></SPAN><SUP><SPAN class=mrow id=id7717165><SPAN class=mi1><EM>n</EM></SPAN><SPAN class=mo id=id7717169>−</SPAN><SPAN class=mn>1</SPAN></SPAN></SUP></SPAN><SPAN class=mo id=id7717174>+</SPAN><SPAN class=mi1><EM>…</EM></SPAN><SPAN class=mo id=id7717179>+</SPAN><SPAN class=mrow id=id7717181><SPAN><SPAN class=mi1><EM>a</EM></SPAN></SPAN><SUB><SPAN class=mn>0</SPAN></SUB></SPAN></SPAN></SPAN><SPAN id=id7717102R>)</SPAN> <SCRIPT type=text/javascript> var mrowH = id7717102M.offsetHeight; mrowStretch(id7717102L,'æ','ç','ç','è'); mrowStretch(id7717102R,'ö','÷','÷','ø');</SCRIPT> <SPAN class=mo id=id7717193></SPAN><SPAN class=mrow id=id7717195><SPAN class=mrow id=id7717196><SPAN><SPAN class=mi1><EM>y</EM></SPAN></SPAN><SUB><SPAN class=mn>0</SPAN></SUB></SPAN><SPAN class=mo id=id7717208></SPAN><SPAN id=id7717210L>(</SPAN><SPAN id=id7717210M><SPAN class=mi1><EM>t</EM></SPAN></SPAN><SPAN id=id7717210R>)</SPAN> <SCRIPT type=text/javascript> var mrowH = id7717210M.offsetHeight; mrowStretch(id7717210L,'æ','ç','ç','è'); mrowStretch(id7717210R,'ö','÷','÷','ø');</SCRIPT> </SPAN></SPAN><SPAN class=mo id=id7717213>=</SPAN><SPAN class=mn>0</SPAN></SPAN></SPAN></SPAN> </FONT> <DIV class=equation id=eq5><FONT size=2><SPAN class=mrow id=id7717154><SPAN class=mrow id=id7717160><SPAN class=mrow id=id7717105><SPAN class=mrow id=id7717106><EM><SPAN class=mi1>c</SPAN><SPAN class=mo id=id7717369></SPAN></EM><SPAN id=id7717370L>(</SPAN><SPAN id=id7717370M><SPAN class=mrow id=id7717376><EM><SPAN><SPAN class=mi1>s</SPAN></SPAN><SUP><SPAN class=mi1>n</SPAN></SUP></EM><SPAN class=mo id=id7717389>+</SPAN><SPAN class=mrow id=id7717391><SPAN class=mrow id=id7717392><SPAN><SPAN class=mi1><EM>a</EM></SPAN></SPAN><SUB><SPAN class=mrow id=id7717400><SPAN class=mi1><EM>n</EM></SPAN><SPAN class=mo id=id7717407>−</SPAN><SPAN class=mn>1</SPAN></SPAN></SUB></SPAN><SPAN class=mo id=id7717416></SPAN><SPAN><SPAN class=mi1><EM>s</EM></SPAN></SPAN><SUP><SPAN class=mrow id=id7717432><SPAN class=mi1><EM>n</EM></SPAN><SPAN class=mo id=id7717436>−</SPAN><SPAN class=mn>1</SPAN></SPAN></SUP></SPAN><SPAN class=mo id=id7717441>+</SPAN><SPAN class=mi1><EM>…</EM></SPAN><SPAN class=mo id=id7717447>+</SPAN><SPAN class=mrow id=id7717449><SPAN class=mrow id=id7717450><SPAN><SPAN class=mi1><EM>a</EM></SPAN></SPAN><SUB><SPAN class=mn>1</SPAN></SUB></SPAN><SPAN class=mo id=id7717462></SPAN><SPAN class=mi1><EM>s</EM></SPAN></SPAN><SPAN class=mo id=id7717466>+</SPAN><SPAN class=mrow id=id7717468><SPAN><SPAN class=mi1><EM>a</EM></SPAN></SPAN><SUB><SPAN class=mn>0</SPAN></SUB></SPAN></SPAN></SPAN><SPAN id=id7717370R>)</SPAN> <SCRIPT type=text/javascript> var mrowH = id7717370M.offsetHeight; mrowStretch(id7717370L,'æ','ç','ç','è'); mrowStretch(id7717370R,'ö','÷','÷','ø');</SCRIPT> <SPAN class=mo id=id7717480></SPAN><EM><SPAN><SPAN class=mi1>ⅇ</SPAN></SPAN><SUP><SPAN class=mrow id=id7717487><SPAN class=mi1>s</SPAN><SPAN class=mo id=id7717491></SPAN><SPAN class=mi1>t</SPAN></SPAN></SUP></EM></SPAN></SPAN><SPAN class=mrow id=id7717494><SPAN class=mo id=id7717495>=</SPAN></SPAN><SPAN class=mrow id=id7717498><SPAN class=mn>0</SPAN></SPAN></SPAN></SPAN> <SPAN class=equation-number>(5)</SPAN> </FONT></DIV><FONT color=#660000><FONT size=2><SPAN class=cnxn-target>equation</SPAN> 5</FONT></FONT><FONT size=2> se mantiene para todo <SPAN class=mrow id=id7717675><SPAN class=mrow id=id7717677><SPAN class=mi1><EM>t</EM></SPAN></SPAN></SPAN> solo cuando </FONT> <DIV class=equation id=eq6><FONT size=2><SPAN class=mrow id=id7717697><SPAN class=mrow id=id7717702><SPAN class=mrow id=id7717709><SPAN class=mrow id=id7717710><EM><SPAN><SPAN class=mi1>s</SPAN></SPAN><SUP><SPAN class=mi1>n</SPAN></SUP></EM><SPAN class=mo id=id7717718>+</SPAN><SPAN class=mrow id=id7717721><SPAN class=mrow id=id7717722><SPAN><SPAN class=mi1><EM>a</EM></SPAN></SPAN><SUB><SPAN class=mrow id=id7717730><SPAN class=mi1><EM>n</EM></SPAN><SPAN class=mo id=id7717736>−</SPAN><SPAN class=mn>1</SPAN></SPAN></SUB></SPAN><SPAN class=mo id=id7717749></SPAN><SPAN><SPAN class=mi1><EM>s</EM></SPAN></SPAN><SUP><SPAN class=mrow id=id7717765><SPAN class=mi1><EM>n</EM></SPAN><SPAN class=mo id=id7717769>−</SPAN><SPAN class=mn>1</SPAN></SPAN></SUP></SPAN><SPAN class=mo id=id7717774>+</SPAN><SPAN class=mi1><EM>…</EM></SPAN><SPAN class=mo id=id7717780>+</SPAN><SPAN class=mrow id=id7717782><SPAN class=mrow id=id7717783><SPAN><SPAN class=mi1><EM>a</EM></SPAN></SPAN><SUB><SPAN class=mn>1</SPAN></SUB></SPAN><SPAN class=mo id=id7717795></SPAN><SPAN class=mi1><EM>s</EM></SPAN></SPAN><SPAN class=mo id=id7717799>+</SPAN><SPAN class=mrow id=id7717801><SPAN><SPAN class=mi1><EM>a</EM></SPAN></SPAN><SUB><SPAN class=mn>0</SPAN></SUB></SPAN></SPAN></SPAN><SPAN class=mrow id=id7717813><SPAN class=mo id=id7717814>=</SPAN></SPAN><SPAN class=mrow id=id7717816><SPAN class=mn>0</SPAN></SPAN></SPAN></SPAN> <SPAN class=equation-number>(6)</SPAN> </FONT></DIV><FONT size=2>Donde esta ecuación es conocida como la <SPAN class=term><STRONG>ecuación característica </STRONG></SPAN>del sistema. Los posibles valores de <SPAN class=mrow id=id7717959><SPAN class=mrow id=id7717964><SPAN class=mi1><EM>s</EM></SPAN></SPAN></SPAN> son las raices o ceros de este polinomio <SPAN class=mrow id=id7717975><SPAN class=mrow id=id7717980><SPAN class=mrow id=id7717982><SPAN id=id7717983L>{</SPAN><SPAN id=id7717983M><SPAN class=mrow id=id7717993><SPAN><SPAN class=mi1><EM>s</EM></SPAN></SPAN><SUB><SPAN class=mn>1</SPAN></SUB></SPAN><SPAN id=id7717993X1>,</SPAN><SPAN class=mrow id=id7718003><SPAN><SPAN class=mi1><EM>s</EM></SPAN></SPAN><SUB><SPAN class=mn>2</SPAN></SUB></SPAN><SPAN id=id7718003X2>,</SPAN><SPAN class=mi1><EM>…</EM></SPAN><SPAN id=id7718013X3>,</SPAN><SPAN class=mrow id=id7718015><EM><SPAN><SPAN class=mi1>s</SPAN></SPAN><SUB><SPAN class=mi1>n</SPAN></SUB></EM></SPAN></SPAN><SPAN id=id7717983R>}</SPAN> <SCRIPT type=text/javascript> var mrowH = id7717983M.offsetHeight; mrowStretch(id7717983L,'ì','ï','í','î'); mrowStretch(id7717983R,'ü','ú','ý','þ');</SCRIPT> </SPAN></SPAN></SPAN><SPAN class=mrow id=id7718079><SPAN class=mrow id=id7718083><SPAN class=mrow id=id7718097><SPAN class=mrow id=id7718098><SPAN id=id7718099L>(</SPAN><SPAN id=id7718099M><SPAN class=mrow id=id7718107><SPAN class=mi1><EM>s</EM></SPAN><SPAN class=mo id=id7718111>−</SPAN><SPAN class=mrow id=id7718114><SPAN><SPAN class=mi1><EM>s</EM></SPAN></SPAN><SUB><SPAN class=mn>1</SPAN></SUB></SPAN></SPAN></SPAN><SPAN id=id7718099R>)</SPAN> <SCRIPT type=text/javascript> var mrowH = id7718099M.offsetHeight; mrowStretch(id7718099L,'æ','ç','ç','è'); mrowStretch(id7718099R,'ö','÷','÷','ø');</SCRIPT> <SPAN class=mo id=id7718126></SPAN><SPAN id=id7718127L>(</SPAN><SPAN id=id7718127M><SPAN class=mrow id=id7718132><SPAN class=mi1><EM>s</EM></SPAN><SPAN class=mo id=id7718136>−</SPAN><SPAN class=mrow id=id7718139><SPAN><SPAN class=mi1><EM>s</EM></SPAN></SPAN><SUB><SPAN class=mn>2</SPAN></SUB></SPAN></SPAN></SPAN><SPAN id=id7718127R>)</SPAN> <SCRIPT type=text/javascript> var mrowH = id7718127M.offsetHeight; mrowStretch(id7718127L,'æ','ç','ç','è'); mrowStretch(id7718127R,'ö','÷','÷','ø');</SCRIPT> <SPAN class=mo id=id7718150></SPAN><SPAN id=id7718151L>(</SPAN><SPAN id=id7718151M><SPAN class=mrow id=id7718156><SPAN class=mi1><EM>s</EM></SPAN><SPAN class=mo id=id7718161>−</SPAN><SPAN class=mrow id=id7718164><SPAN><SPAN class=mi1><EM>s</EM></SPAN></SPAN><SUB><SPAN class=mn>3</SPAN></SUB></SPAN></SPAN></SPAN><SPAN id=id7718151R>)</SPAN> <SCRIPT type=text/javascript> var mrowH = id7718151M.offsetHeight; mrowStretch(id7718151L,'æ','ç','ç','è'); mrowStretch(id7718151R,'ö','÷','÷','ø');</SCRIPT> <SPAN class=mo id=id7718175></SPAN><EM><SPAN class=mi1>…</SPAN><SPAN class=mo id=id7718178></SPAN></EM><SPAN id=id7718180L>(</SPAN><SPAN id=id7718180M><SPAN class=mrow id=id7718185><SPAN class=mi1><EM>s</EM></SPAN><SPAN class=mo id=id7718189>−</SPAN><SPAN class=mrow id=id7718192><EM><SPAN><SPAN class=mi1>s</SPAN></SPAN><SUB><SPAN class=mi1>n</SPAN></SUB></EM></SPAN></SPAN></SPAN><SPAN id=id7718180R>)</SPAN> <SCRIPT type=text/javascript> var mrowH = id7718180M.offsetHeight; mrowStretch(id7718180L,'æ','ç','ç','è'); mrowStretch(id7718180R,'ö','÷','÷','ø');</SCRIPT> </SPAN><SPAN class=mo id=id7718204>=</SPAN><SPAN class=mn>0</SPAN></SPAN></SPAN></SPAN> es decir las posibles soluciones son: <SPAN class=mrow id=id7718330><SPAN class=mrow id=id7718089><SPAN class=mrow id=id7718332><SPAN class=mrow id=id7718333><SPAN><SPAN class=mi1><EM>c</EM></SPAN></SPAN><SUB><SPAN class=mn>1</SPAN></SUB></SPAN><SPAN class=mo id=id7718345></SPAN><SPAN><SPAN class=mi1><EM>ⅇ</EM></SPAN></SPAN><SUP><SPAN class=mrow id=id7718359><SPAN class=mrow id=id7718360><SPAN><SPAN class=mi1><EM>s</EM></SPAN></SPAN><SUB><SPAN class=mn>1</SPAN></SUB></SPAN><SPAN class=mo id=id7718371></SPAN><SPAN class=mi1><EM>t</EM></SPAN></SPAN></SUP></SPAN></SPAN></SPAN>, <SPAN class=mrow id=id7718429><SPAN class=mrow id=id7718356><SPAN class=mrow id=id7718432><SPAN class=mrow id=id7718433><SPAN><SPAN class=mi1><EM>c</EM></SPAN></SPAN><SUB><SPAN class=mn>2</SPAN></SUB></SPAN><SPAN class=mo id=id7718444></SPAN><SPAN><SPAN class=mi1><EM>ⅇ</EM></SPAN></SPAN><SUP><SPAN class=mrow id=id7718458><SPAN class=mrow id=id7718459><SPAN><SPAN class=mi1><EM>s</EM></SPAN></SPAN><SUB><SPAN class=mn>2</SPAN></SUB></SPAN><SPAN class=mo id=id7718470></SPAN><SPAN class=mi1><EM>t</EM></SPAN></SPAN></SUP></SPAN></SPAN></SPAN>, <SPAN class=mrow id=id7718528><SPAN class=mrow id=id7718455><SPAN class=mi1><EM>…</EM></SPAN></SPAN></SPAN>, <SPAN class=mrow id=id7718536><SPAN class=mrow id=id7718540><SPAN class=mrow id=id7718549><EM><SPAN class=mrow id=id7718550><SPAN><SPAN class=mi1>c</SPAN></SPAN><SUB><SPAN class=mi1>n</SPAN></SUB></SPAN><SPAN class=mo id=id7718562></SPAN><SPAN><SPAN class=mi1>ⅇ</SPAN></SPAN><SUP><SPAN class=mrow id=id7718575><SPAN class=mrow id=id7718576><SPAN><SPAN class=mi1>s</SPAN></SPAN><SUB><SPAN class=mi1>n</SPAN></SUB></SPAN><SPAN class=mo id=id7718588></SPAN><SPAN class=mi1>t</SPAN></SPAN></SUP></EM></SPAN></SPAN></SPAN>. Ya que el sistema es <I>lineal</I>, la solución general es de la forma: </FONT> <DIV class=equation id=eq7><FONT size=2><SPAN class=mrow id=id7718565><SPAN class=mrow id=id7718547><SPAN class=mrow id=id7718670><SPAN class=mrow id=id7718671><SPAN class=mrow id=id7718672><SPAN><SPAN class=mi1><EM>y</EM></SPAN></SPAN><SUB><SPAN class=mn>0</SPAN></SUB></SPAN><SPAN class=mo id=id7718683></SPAN><SPAN id=id7718685L>(</SPAN><SPAN id=id7718685M><SPAN class=mi1><EM>t</EM></SPAN></SPAN><SPAN id=id7718685R>)</SPAN> <SCRIPT type=text/javascript> var mrowH = id7718685M.offsetHeight; mrowStretch(id7718685L,'æ','ç','ç','è'); mrowStretch(id7718685R,'ö','÷','÷','ø');</SCRIPT> </SPAN></SPAN><SPAN class=mrow id=id7718689><SPAN class=mo id=id7718690>=</SPAN></SPAN><SPAN class=mrow id=id7718692><SPAN class=mrow id=id7718695><SPAN class=mrow id=id7718696><SPAN class=mrow id=id7718697><SPAN><SPAN class=mi1><EM>c</EM></SPAN></SPAN><SUB><SPAN class=mn>1</SPAN></SUB></SPAN><SPAN class=mo id=id7718709></SPAN><SPAN><SPAN class=mi1><EM>ⅇ</EM></SPAN></SPAN><SUP><SPAN class=mrow id=id7718722><SPAN class=mrow id=id7718723><SPAN><SPAN class=mi1><EM>s</EM></SPAN></SPAN><SUB><SPAN class=mn>1</SPAN></SUB></SPAN><SPAN class=mo id=id7718734></SPAN><SPAN class=mi1><EM>t</EM></SPAN></SPAN></SUP></SPAN><SPAN class=mo id=id7718738>+</SPAN><SPAN class=mrow id=id7718741><SPAN class=mrow id=id7718742><SPAN><SPAN class=mi1><EM>c</EM></SPAN></SPAN><SUB><SPAN class=mn>2</SPAN></SUB></SPAN><SPAN class=mo id=id7718754></SPAN><SPAN><SPAN class=mi1><EM>ⅇ</EM></SPAN></SPAN><SUP><SPAN class=mrow id=id7718763><SPAN class=mrow id=id7718764><SPAN><SPAN class=mi1><EM>s</EM></SPAN></SPAN><SUB><SPAN class=mn>2</SPAN></SUB></SPAN><SPAN class=mo id=id7718776></SPAN><SPAN class=mi1><EM>t</EM></SPAN></SPAN></SUP></SPAN><SPAN class=mo id=id7718780>+</SPAN><SPAN class=mi1><EM>…</EM></SPAN><SPAN class=mo id=id7718785>+</SPAN><SPAN class=mrow id=id7718788><EM><SPAN class=mrow id=id7718789><SPAN><SPAN class=mi1>c</SPAN></SPAN><SUB><SPAN class=mi1>n</SPAN></SUB></SPAN><SPAN class=mo id=id7718800></SPAN><SPAN><SPAN class=mi1>ⅇ</SPAN></SPAN><SUP><SPAN class=mrow id=id7718808><SPAN class=mrow id=id7718809><SPAN><SPAN class=mi1>s</SPAN></SPAN><SUB><SPAN class=mi1>n</SPAN></SUB></SPAN><SPAN class=mo id=id7718821></SPAN><SPAN class=mi1>t</SPAN></SPAN></SUP></EM></SPAN></SPAN></SPAN></SPAN></SPAN> <SPAN class=equation-number>(7)</SPAN> </FONT></DIV><FONT size=2>Entonces, resolver para <SPAN class=mrow id=id7718666><SPAN class=mrow id=id7718719><SPAN class=mrow id=id7718660><SPAN id=id7719025L>{</SPAN><SPAN id=id7719025M><SPAN class=mrow id=id7719033><SPAN><SPAN class=mi1><EM>c</EM></SPAN></SPAN><SUB><SPAN class=mn>1</SPAN></SUB></SPAN><SPAN id=id7719033X1>,</SPAN><SPAN class=mi1><EM>…</EM></SPAN><SPAN id=id7719047X2>,</SPAN><SPAN class=mrow id=id7719050><EM><SPAN><SPAN class=mi1>c</SPAN></SPAN><SUB><SPAN class=mi1>n</SPAN></SUB></EM></SPAN></SPAN><SPAN id=id7719025R>}</SPAN> <SCRIPT type=text/javascript> var mrowH = id7719025M.offsetHeight; mrowStretch(id7719025L,'ì','ï','í','î'); mrowStretch(id7719025R,'ü','ú','ý','þ');</SCRIPT> </SPAN></SPAN></SPAN>usando las condiciones iniciales. </FONT></DIV> <DIV class=example id=eg_opt><FONT size=2><STRONG><SPAN class=example-before><FONT color=#336699>Example 3</FONT></SPAN><SPAN class=example-name> </SPAN></STRONG> </FONT> <DIV class=para id=p1_egopt><FONT size=2>Véase Lathi p.108 para un buen ejemplo. </FONT></DIV></DIV> <DIV class=para id=pf_sec2><FONT size=2>Generalmente asumimos que las IC's de un sistema son cero, lo que implica que <SPAN class=mrow id=id7719125><SPAN class=mrow id=id7719129><SPAN class=mrow id=id7719147><SPAN class=mrow id=id7719148><EM><SPAN class=mrow id=id7719149><SPAN><SPAN class=mi1>y</SPAN></SPAN><SUB><SPAN class=mi1>i</SPAN></SUB></SPAN><SPAN class=mo id=id7719160></SPAN></EM><SPAN id=id7719162L>(</SPAN><SPAN id=id7719162M><SPAN class=mi1><EM>t</EM></SPAN></SPAN><SPAN id=id7719162R>)</SPAN> <SCRIPT type=text/javascript> var mrowH = id7719162M.offsetHeight; mrowStretch(id7719162L,'æ','ç','ç','è'); mrowStretch(id7719162R,'ö','÷','÷','ø');</SCRIPT> </SPAN><SPAN class=mo id=id7719166>=</SPAN><SPAN class=mn>0</SPAN></SPAN></SPAN></SPAN>. Sin embargo, el<I>método</I> de resolver para <SPAN class=mrow id=id7719223><SPAN class=mrow id=id7719136><SPAN class=mrow id=id7719229><EM><SPAN class=mrow id=id7719230><SPAN><SPAN class=mi1>y</SPAN></SPAN><SUB><SPAN class=mi1>i</SPAN></SUB></SPAN><SPAN class=mo id=id7719240></SPAN></EM><SPAN id=id7719243L>(</SPAN><SPAN id=id7719243M><SPAN class=mi1><EM>t</EM></SPAN></SPAN><SPAN id=id7719243R>)</SPAN> <SCRIPT type=text/javascript> var mrowH = id7719243M.offsetHeight; mrowStretch(id7719243L,'æ','ç','ç','è'); mrowStretch(id7719243R,'ö','÷','÷','ø');</SCRIPT> </SPAN></SPAN></SPAN>se probará más adelante. </FONT></DIV></DIV> <DIV class=section id=sec3> <H2><FONT size=2>Encontrando la Respuesta del Estado-Cero</FONT></H2> <DIV class=para id=p1_s3><FONT size=2>Resolviendo una ecuación lienal diferencial </FONT> <DIV class=equation id=eq10><SPAN class=mrow id=id7719300><SPAN class=mrow id=id7719304><SPAN class=mrow id=id7719318><SPAN class=mrow id=id7719319> <TABLE class=munderover> <TBODY> <TR> <TD><SPAN class=mrow id=id7719335><SPAN class=mi1><EM><FONT size=2>n</FONT></EM></SPAN></SPAN></TD></TR> <TR> <TD><SPAN class=mo id=id7719322><FONT size=2>∑</FONT></SPAN></TD></TR> <TR> <TD><SPAN class=mrow id=id7719324><FONT size=2><SPAN class=mi1><EM>i</EM></SPAN><SPAN class=mo id=id7719329>=</SPAN><SPAN class=mn>0</SPAN></FONT></SPAN></TD></TR></TBODY></TABLE><SPAN id=id7719340L> <TABLE class=lr> <TBODY> <TR> <TD><FONT size=2><FONT face=symbol>æ</FONT><BR><FONT face=symbol>ç</FONT><BR><FONT face=symbol>è</FONT></FONT></TD></TR></TBODY></TABLE></SPAN><SPAN id=id7719340M><SPAN class=mrow id=id7719344><EM><FONT size=2><SPAN class=mrow id=id7719345><SPAN><SPAN class=mi1>a</SPAN></SPAN><SUB><SPAN class=mi1>i</SPAN></SUB></SPAN><SPAN class=mo id=id7719356></SPAN></FONT></EM><SPAN class=mrow id=id7719363><FONT size=2> </FONT> <TABLE class=mfrac style="FONT-SIZE: 75%"> <TBODY> <TR class=mfraca id=aid7719364> <TD class=mfracaa><SPAN class=mrow id=id7719365><FONT size=2><SPAN><SPAN class=mo id=id7719367>d</SPAN></SPAN><SUP><SPAN class=mrow id=id7719369><SPAN class=mi1><EM>i</EM></SPAN></SPAN></SUP></FONT></SPAN></TD></TR> <TR class=mfracb id=bid7719364 style="HEIGHT: 22px"> <TD><SPAN class=mrow id=id7719375><FONT size=2><SPAN class=mo id=id7719376>d</SPAN><EM><SPAN><SPAN class=mrow id=id7719379><SPAN class=mi1>t</SPAN></SPAN></SPAN><SUP><SPAN class=mrow id=id7719384><SPAN class=mi1>i</SPAN></SPAN></SUP></EM></FONT></SPAN></TD></TR></TBODY></TABLE> <SCRIPT type=text/javascript> if ( aid7719364.offsetHeight > bid7719364.offsetHeight ) bid7719364.style.setExpression("height",aid7719364.offsetHeight ); else aid7719364.style.setExpression("height",bid7719364.offsetHeight ); </SCRIPT> <SPAN class=mrow id=id7719388><SPAN class=mrow id=id7719394><FONT size=2><EM><SPAN class=mi1>y</SPAN><SPAN class=mo id=id7719397></SPAN></EM><SPAN id=id7719399L>(</SPAN><SPAN id=id7719399M><SPAN class=mi1><EM>t</EM></SPAN></SPAN><SPAN id=id7719399R>)</SPAN> <SCRIPT type=text/javascript> var mrowH = id7719399M.offsetHeight; mrowStretch(id7719399L,'æ','ç','ç','è'); mrowStretch(id7719399R,'ö','÷','÷','ø');</SCRIPT> </FONT></SPAN></SPAN></SPAN></SPAN></SPAN><SPAN id=id7719340R> <TABLE class=lr> <TBODY> <TR> <TD><FONT size=2><FONT face=symbol>ö</FONT><BR><FONT face=symbol>÷</FONT><BR><FONT face=symbol>ø</FONT></FONT></TD></TR></TBODY></TABLE></SPAN> <SCRIPT type=text/javascript> var mrowH = id7719340M.offsetHeight; mrowStretch(id7719340L,'æ','ç','ç','è'); mrowStretch(id7719340R,'ö','÷','÷','ø');</SCRIPT> </SPAN></SPAN><SPAN class=mrow id=id7719402><SPAN class=mo id=id7719403><FONT size=2>=</FONT></SPAN></SPAN><SPAN class=mrow id=id7719406><SPAN class=mrow id=id7719407><FONT size=2> </FONT> <TABLE class=munderover> <TBODY> <TR> <TD><SPAN class=mrow id=id7719422><SPAN class=mi1><EM><FONT size=2>m</FONT></EM></SPAN></SPAN></TD></TR> <TR> <TD><SPAN class=mo id=id7719410><FONT size=2>∑</FONT></SPAN></TD></TR> <TR> <TD><SPAN class=mrow id=id7719412><FONT size=2><SPAN class=mi1><EM>i</EM></SPAN><SPAN class=mo id=id7719417>=</SPAN><SPAN class=mn>0</SPAN></FONT></SPAN></TD></TR></TBODY></TABLE><SPAN id=id7719427L> <TABLE class=lr> <TBODY> <TR> <TD><FONT size=2><FONT face=symbol>æ</FONT><BR><FONT face=symbol>ç</FONT><BR><FONT face=symbol>è</FONT></FONT></TD></TR></TBODY></TABLE></SPAN><SPAN id=id7719427M><SPAN class=mrow id=id7719430><EM><FONT size=2><SPAN class=mrow id=id7719431><SPAN><SPAN class=mi1>b</SPAN></SPAN><SUB><SPAN class=mi1>i</SPAN></SUB></SPAN><SPAN class=mo id=id7719443></SPAN></FONT></EM><SPAN class=mrow id=id7719446><FONT size=2> </FONT> <TABLE class=mfrac style="FONT-SIZE: 75%"> <TBODY> <TR class=mfraca id=aid7719447> <TD class=mfracaa><SPAN class=mrow id=id7719448><FONT size=2><SPAN><SPAN class=mo id=id7719450>d</SPAN></SPAN><SUP><SPAN class=mrow id=id7719453><SPAN class=mi1><EM>i</EM></SPAN></SPAN></SUP></FONT></SPAN></TD></TR> <TR class=mfracb id=bid7719447 style="HEIGHT: 22px"> <TD><SPAN class=mrow id=id7719457><FONT size=2><SPAN class=mo id=id7719458>d</SPAN><EM><SPAN><SPAN class=mrow id=id7719462><SPAN class=mi1>t</SPAN></SPAN></SPAN><SUP><SPAN class=mrow id=id7719467><SPAN class=mi1>i</SPAN></SPAN></SUP></EM></FONT></SPAN></TD></TR></TBODY></TABLE> <SCRIPT type=text/javascript> if ( aid7719447.offsetHeight > bid7719447.offsetHeight ) bid7719447.style.setExpression("height",aid7719447.offsetHeight ); else aid7719447.style.setExpression("height",bid7719447.offsetHeight ); </SCRIPT> <SPAN class=mrow id=id7719470><SPAN class=mrow id=id7719473><FONT size=2><EM><SPAN class=mi1>f</SPAN><SPAN class=mo id=id7719476></SPAN></EM><SPAN id=id7719478L>(</SPAN><SPAN id=id7719478M><SPAN class=mi1><EM>t</EM></SPAN></SPAN><SPAN id=id7719478R>)</SPAN> <SCRIPT type=text/javascript> var mrowH = id7719478M.offsetHeight; mrowStretch(id7719478L,'æ','ç','ç','è'); mrowStretch(id7719478R,'ö','÷','÷','ø');</SCRIPT> </FONT></SPAN></SPAN></SPAN></SPAN></SPAN><SPAN id=id7719427R> <TABLE class=lr> <TBODY> <TR> <TD><FONT size=2><FONT face=symbol>ö</FONT><BR><FONT face=symbol>÷</FONT><BR><FONT face=symbol>ø</FONT></FONT></TD></TR></TBODY></TABLE></SPAN> <SCRIPT type=text/javascript> var mrowH = id7719427M.offsetHeight; mrowStretch(id7719427L,'æ','ç','ç','è'); mrowStretch(id7719427R,'ö','÷','÷','ø');</SCRIPT> </SPAN></SPAN></SPAN></SPAN><FONT size=2><SPAN class=equation-number>(8)</SPAN> </FONT></DIV><FONT size=2>dada una entrada específica <SPAN class=mrow id=id7719310><SPAN class=mrow id=id7719360><SPAN class=mrow id=id7719669><EM><SPAN class=mi1>f</SPAN><SPAN class=mo id=id7719672></SPAN></EM><SPAN id=id7719674L>(</SPAN><SPAN id=id7719674M><SPAN class=mi1><EM>t</EM></SPAN></SPAN><SPAN id=id7719674R>)</SPAN> <SCRIPT type=text/javascript> var mrowH = id7719674M.offsetHeight; mrowStretch(id7719674L,'æ','ç','ç','è'); mrowStretch(id7719674R,'ö','÷','÷','ø');</SCRIPT> </SPAN></SPAN></SPAN>es una tarea difícil en general. Más importantemente, el método depende completamente en la naturaleza de <SPAN class=mrow id=id7719697><SPAN class=mrow id=id7719700><SPAN class=mrow id=id7719705><EM><SPAN class=mi1>f</SPAN><SPAN class=mo id=id7719709></SPAN></EM><SPAN id=id7719711L>(</SPAN><SPAN id=id7719711M><SPAN class=mi1><EM>t</EM></SPAN></SPAN><SPAN id=id7719711R>)</SPAN> <SCRIPT type=text/javascript> var mrowH = id7719711M.offsetHeight; mrowStretch(id7719711L,'æ','ç','ç','è'); mrowStretch(id7719711R,'ö','÷','÷','ø');</SCRIPT> </SPAN></SPAN></SPAN>; si cambiamos la señal de entrada, debemos de re-resolver el sistema de ecuaciones para encontrar la respuesta del sistema. </FONT></DIV> <DIV class=para id=p2_s3><FONT size=2>La </FONT><FONT color=#660000 size=2>Convolución</FONT><FONT size=2> nos ayuda a pasar estas dificultades. En la sección explicamos como la convolución nos ayuda a determinar la salida del sistema, dada solo la entrada <SPAN class=mrow id=id7719751><SPAN class=mrow id=id7719754><SPAN class=mrow id=id7719763><EM><SPAN class=mi1>f</SPAN><SPAN class=mo id=id7719766></SPAN></EM><SPAN id=id7719768L>(</SPAN><SPAN id=id7719768M><SPAN class=mi1><EM>t</EM></SPAN></SPAN><SPAN id=id7719768R>)</SPAN> <SCRIPT type=text/javascript> var mrowH = id7719768M.offsetHeight; mrowStretch(id7719768L,'æ','ç','ç','è'); mrowStretch(id7719768R,'ö','÷','÷','ø');</SCRIPT> </SPAN></SPAN></SPAN>y la </FONT><FONT color=#660000 size=2>respuesta al impulso </FONT><FONT size=2>del sistema, <SPAN class=mrow id=id7719800><SPAN class=mrow id=id7719801><SPAN class=mrow id=id7719808><EM><SPAN class=mi1>h</SPAN><SPAN class=mo id=id7719812></SPAN></EM><SPAN id=id7719814L>(</SPAN><SPAN id=id7719814M><SPAN class=mi1><EM>t</EM></SPAN></SPAN><SPAN id=id7719814R>)</SPAN> <SCRIPT type=text/javascript> var mrowH = id7719814M.off