document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1184644>Question 1184644</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve the initial-value problem d^(2)y/dt^(2)+4dy/dt+4y=0,y(1)=0,y′(1)=1.\r<BR>\n" );
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document.write( "y(t)= </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #815294 by </B><A HREF=/tutors/aboutme.mpl?userid=robertb><B>robertb(5830)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=robertb><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=robertb><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=815294\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <BR>\n" );
document.write( "The characteristic equation of the D.E. is  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=lambda%5E2+%2B+4%2Alambda+%2B+4+=+0 ALIGN=MIDDLE  ALT=\"lambda%5E2+%2B+4%2Alambda+%2B+4+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, which gives the double root <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=lambda+=+-2 ALIGN=MIDDLE  ALT=\"lambda+=+-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "===>  The general solution to the D.E. is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+A%2Ae%5E%28-2t%29+%2B+Bt%2Ae%5E%28-2t%29 ALIGN=MIDDLE  ALT=\"y+=+A%2Ae%5E%28-2t%29+%2B+Bt%2Ae%5E%28-2t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22y%27%281%29%22+=+%28-2A%2BB%29e%5E%28-2%29+-+2Be%5E%28-2%29+=1 ALIGN=MIDDLE  ALT=\"%22y%27%281%29%22+=+%28-2A%2BB%29e%5E%28-2%29+-+2Be%5E%28-2%29+=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > ===> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-2A-B%29e%5E%28-2%29+=+1 ALIGN=MIDDLE  ALT=\"%28-2A-B%29e%5E%28-2%29+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" > ===> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2A%2BB+=+-e%5E2 ALIGN=MIDDLE  ALT=\"2A%2BB+=+-e%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Also, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%281%29+=+Ae%5E%28-2%29+%2B+Be%5E%28-2%29+=+0 ALIGN=MIDDLE  ALT=\"y%281%29+=+Ae%5E%28-2%29+%2B+Be%5E%28-2%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ===> A + B = 0, or B = -A.\r<BR>\n" );
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document.write( "Combining this with the preceding equation, we get \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2A+-+A+=+A+=+-e%5E2 ALIGN=MIDDLE  ALT=\"2A+-+A+=+A+=+-e%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.   ===> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B+=e%5E2 ALIGN=MIDDLE  ALT=\"B+=e%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >   ===> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+-e%5E2%2Ae%5E%28-2t%29+%2B+e%5E2%2At%2Ae%5E%28-2t%29 ALIGN=MIDDLE  ALT=\"y+=+-e%5E2%2Ae%5E%28-2t%29+%2B+e%5E2%2At%2Ae%5E%28-2t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "after combining factors and simplifying, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%28t%29+=+%28t-1%29e%5E%28-2%28t-1%29%29 ALIGN=MIDDLE  ALT=\"y%28t%29+=+%28t-1%29e%5E%28-2%28t-1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " </SPAN>\n" );
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