document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=815508>Question 815508</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The true solution at x=0.5 for  y'= -xy at y(0)=1 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #491117 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=491117\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> y'= -xy\r<BR>\n" );
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document.write( "dy/dx= -xy\r<BR>\n" );
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document.write( "dy= -xy*dx\r<BR>\n" );
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document.write( "dy/y= -x*dx\r<BR>\n" );
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document.write( "int( dy/y )= int( -x*dx ) ... integrating both sides\r<BR>\n" );
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document.write( "ln( |y| ) = (-1/2)*x^2 + C ... don't forget the \" + C \" part\r<BR>\n" );
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document.write( "|y| = e^[(-1/2)*x^2 + C]\r<BR>\n" );
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document.write( "|y| = e^((-1/2)*x^2)*(e^C)\r<BR>\n" );
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document.write( "|y| = e^C * e^((-1/2)*x^2)\r<BR>\n" );
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document.write( "y = +-e^C * e^((-1/2)*x^2)\r<BR>\n" );
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document.write( "Because e^C and e^((-1/2)*x^2) are NEVER negative, this means that the expression e^C * e^((-1/2)*x^2) is NEVER negative.\r<BR>\n" );
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document.write( "So that fact, along with the fact that y(0) = 1, which is positive, means that we're going to focus on the \"plus\" and ignore the \"minus\"\r<BR>\n" );
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document.write( "Basically, we now have y = e^C * e^((-1/2)*x^2)\r<BR>\n" );
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document.write( "Plug in x = 0 and y = 1, from the initial condition y(0) = 1, then solve for C\r<BR>\n" );
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document.write( "y = e^C * e^((-1/2)*x^2)\r<BR>\n" );
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document.write( "1 = e^C * e^((-1/2)*0^2)\r<BR>\n" );
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document.write( "1 = e^C * e^((-1/2)*0)\r<BR>\n" );
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document.write( "1 = e^C * e^(0)\r<BR>\n" );
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document.write( "1 = e^C * 1\r<BR>\n" );
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document.write( "1 = e^C\r<BR>\n" );
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document.write( "e^C = 1\r<BR>\n" );
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document.write( "C = ln(1)\r<BR>\n" );
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document.write( "C = 0\r<BR>\n" );
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document.write( "So\r<BR>\n" );
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document.write( "y = e^C * e^((-1/2)*x^2)\r<BR>\n" );
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document.write( "turns into\r<BR>\n" );
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document.write( "y = e^0 * e^((-1/2)*x^2)\r<BR>\n" );
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document.write( "y = 1 * e^((-1/2)*x^2)\r<BR>\n" );
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document.write( "y = e^((-1/2)*x^2)\r<BR>\n" );
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document.write( "which is the solution that satisfies both the differential equation y' = -xy and the initial condition y(0) = 1\r<BR>\n" );
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document.write( "To find the value of y at x = 0.5, plug it in and evaluate.\r<BR>\n" );
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document.write( "y = e^((-1/2)*x^2)\r<BR>\n" );
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document.write( "y = e^((-1/2)*(0.5)^2)\r<BR>\n" );
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document.write( "y = e^((-1/2)*0.25)\r<BR>\n" );
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document.write( "y = e^(-0.125)\r<BR>\n" );
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document.write( "y = 0.88249690258456 (this is approximate)\r<BR>\n" );
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document.write( "So when x = 0.5, y is approximately <font color=\"red\">y = 0.88249690258456</font>\r<BR>\n" );
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