document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1184041>Question 1184041</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Use the \"mixed partials\" check to see if the following differential equation is exact.\r<BR>\n" );
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document.write( "If it is exact find a function F(x,y) whose differential, dF(x,y) gives the differential equation. That is, level curves F(x,y)=C are solutions to the differential equation:\r<BR>\n" );
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document.write( "dy/dx=−x^4−2y/2x+4y^2\r<BR>\n" );
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document.write( "First rewrite as M(x,y)dx+N(x,y)dy=0\r<BR>\n" );
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document.write( "where M(x,y)= My answer is x^4-2y but it's wrong.<BR>\n" );
document.write( "and N(x,y)= My answer is -2x+4y^2 (wrong)\r<BR>\n" );
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document.write( "If the equation is not exact, enter not exact, otherwise enter in F(x,y) as the solution of the differential equation here: \r<BR>\n" );
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document.write( "My answer is -(x^5/5)-2yx-(4/3)y^3=C but it's wrong.\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #814584 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=814584\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Next time, try to make proper use of parentheses. I will assume that your D.E. is\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dy%2Fdx+=+%28-x%5E4+-2y%29%2F%282x+%2B+4y%5E2%29 ALIGN=MIDDLE  ALT=\"dy%2Fdx+=+%28-x%5E4+-2y%29%2F%282x+%2B+4y%5E2%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=%28x%5E4+%2B2y%29dx+%2B+%282x+%2B+4y%5E2%29dy=0 ALIGN=MIDDLE  ALT=\"%28x%5E4+%2B2y%29dx+%2B+%282x+%2B+4y%5E2%29dy=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=M%28x%2Cy%29+=+x%5E4+%2B2y ALIGN=MIDDLE  ALT=\"M%28x%2Cy%29+=+x%5E4+%2B2y\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=N%28x%2Cy%29+=2x+%2B+4y%5E2 ALIGN=MIDDLE  ALT=\"N%28x%2Cy%29+=2x+%2B+4y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "As <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=M%5By%5D+=+2+=+N%5Bx%5D ALIGN=MIDDLE  ALT=\"M%5By%5D+=+2+=+N%5Bx%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the D.E. is exact. \r<BR>\n" );
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document.write( "As such, you can retrieve the potential function F(x,y) by setting ∂F/∂x = M(x,y) and ∂F/∂y = N(x,y).\r<BR>\n" );
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document.write( "In the end what you will get is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=F%28x%2Cy%29+=+x%5E5%2F5+%2B+2yx+%2B+%284%2F3%29y%5E3+=+C+ ALIGN=MIDDLE  ALT=\"F%28x%2Cy%29+=+x%5E5%2F5+%2B+2yx+%2B+%284%2F3%29y%5E3+=+C+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Can you see now where you went wrong?\r<BR>\n" );
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