document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=456139>Question 456139</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Obtain the general solution\r<BR>\n" );
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document.write( "y lnx lny dx + dy = 0 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #313101 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=313101\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Divide both sides by ylny, to get\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=lnx+dx+%2B+dy%2F%28ylny%29+=+0 ALIGN=MIDDLE  ALT=\"lnx+dx+%2B+dy%2F%28ylny%29+=+0\"   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=lnx+dx+%2B+d%28lny%29%2Flny+=+0 ALIGN=MIDDLE  ALT=\"lnx+dx+%2B+d%28lny%29%2Flny+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Integrate both sides:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=int%28lnx%2C+dx%29+%2B+int%281%2Flny%2C+d%28lny%29%29+=+k ALIGN=MIDDLE  ALT=\"int%28lnx%2C+dx%29+%2B+int%281%2Flny%2C+d%28lny%29%29+=+k\"   DISABLED_style= \"max-width:90%; height:auto;\" > k an arbitrary constant.\r<BR>\n" );
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document.write( "After integration by parts (on the first integral), the equation becomes \r<BR>\n" );
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document.write( "xlnx - x + ln(lny) = k.\r<BR>\n" );
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document.write( "Simplifying:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%28x%5Ex%29+%2B+ln%28lny%29+=+x%2Bk ALIGN=MIDDLE  ALT=\"ln%28x%5Ex%29+%2B+ln%28lny%29+=+x%2Bk\"   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=ln%28yx%5Ex%29+=+x%2Bk ALIGN=MIDDLE  ALT=\"ln%28yx%5Ex%29+=+x%2Bk\"   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=yx%5Ex+=+e%5E%28x%2Bk%29 ALIGN=MIDDLE  ALT=\"yx%5Ex+=+e%5E%28x%2Bk%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=yx%5Ex+=+Ce%5Ex ALIGN=MIDDLE  ALT=\"yx%5Ex+=+Ce%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" > <==>  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+C%28e%2Fx%29%5Ex ALIGN=MIDDLE  ALT=\"y+=+C%28e%2Fx%29%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" >, where C is an arbitrary constant. </SPAN>\n" );
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