document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=798503>Question 798503</A>: <I><SPAN STYLE=\"word-break: break-all;\"> This question relates to finding partial derivatives.\r<BR>\n" );
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document.write( "Burger's Equation is a partial differential equation, used for describing wave processes in acoustics and hydrodynamics,<BR>\n" );
document.write( "(let d= partial derivative, a=alpha, and L=lambda)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28dw%2Fdt%29=+%28%28d%5E2w%2Fdx%5E2%29%2B+w%28dw%2Fdx%29%29%29 ALIGN=MIDDLE  ALT=\"+%28dw%2Fdt%29=+%28%28d%5E2w%2Fdx%5E2%29%2B+w%28dw%2Fdx%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "verify that<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=w%28x%2Ct%29=+L+%2B+%282%2F%28x%2BLt%2B+a%29%29 ALIGN=MIDDLE  ALT=\"w%28x%2Ct%29=+L+%2B+%282%2F%28x%2BLt%2B+a%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "is a solution, where L and a are arbitary constants\r<BR>\n" );
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document.write( "thank-you for any help  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #482296 by </B><A HREF=/tutors/aboutme.mpl?userid=rothauserc><B>rothauserc(4718)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=rothauserc><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=rothauserc><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=482296\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> We are going to take two partial derivatives of the equation <BR>\n" );
document.write( "w(x, t) = L + (2 / (x +Lt +a)<BR>\n" );
document.write( "rewite the equation as<BR>\n" );
document.write( "w(x, t) = L + 2*(x +Lt +a)-1<BR>\n" );
document.write( "note that L and a are arbitrary constants<BR>\n" );
document.write( "@w/@t = 2*-1*L*(x +Lt +a)-2 = -2L/(x +Lt +a)^2<BR>\n" );
document.write( "note that for the above partial derivative x is constant<BR>\n" );
document.write( "@w/@x = 2*-1*(x +Lt +a)^-2 = -2 / (x +Lt +a)^2<BR>\n" );
document.write( "note that for the above partial derivative t is constant<BR>\n" );
document.write( "@w/@x ----> @^2w/@x^2 = @w/@x (-2 * (x +Lt +a)^-2 = 4 / (x +Lt +a)^3<BR>\n" );
document.write( "now we can calculate<BR>\n" );
document.write( "@^2w/@x^2 + w*@w/@X<BR>\n" );
document.write( "= 4 / (x +Lt +a)^3 + (L +2 / (x +Lt +a)) * (-2 / (x +Lt +a)^2)<BR>\n" );
document.write( "= 4 / (x +Lt +a)^3 + -2L / (x +Lt +a)^2 -4 / (x +Lt +a)^3<BR>\n" );
document.write( "= -2L / (x +Lt +a)^2<BR>\n" );
document.write( "= @w/@t, as required.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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