document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1115567>Question 1115567</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help me answer the following question:\r<BR>\n" );
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document.write( "(a) Determine the particular solution of the equation\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+d%5E2y%2Fdx%5E2+-+3%28dy%2Fdx%29+=+9+ ALIGN=MIDDLE  ALT=\"+d%5E2y%2Fdx%5E2+-+3%28dy%2Fdx%29+=+9+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "given the initial conditions:<BR>\n" );
document.write( "y(0) = 0,     y'(0) = 0 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #730477 by </B><A HREF=/tutors/aboutme.mpl?userid=math_helper><B>math_helper(2461)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_helper><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=math_helper><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=730477\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Using Laplace transforms: <br>\r<BR>\n" );
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document.write( " L(y'') = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+s%5E2+ ALIGN=MIDDLE  ALT=\"+s%5E2+\"   DISABLED_style= \"max-width:90%; height:auto;\" >Y(s) - sy(0) - y'(0) <BR>\n" );
document.write( " L(y') =  sY(s) - y(0)<BR>\n" );
document.write( " L(c) =  c/s      (c=a constant) <br>\r<BR>\n" );
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document.write( " Noting y'(0)=y(0)=0, the Laplace transform is:\r<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=++s%5E2Y%28s%29+-+3sY%28s%29+=+9%2Fs+ ALIGN=MIDDLE  ALT=\"++s%5E2Y%28s%29+-+3sY%28s%29+=+9%2Fs+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br>\r<BR>\n" );
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document.write( "Now we \"just\" need to isolate Y(s) and then take the inverse Laplace transform.<BR>\n" );
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document.write( "   <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+Y%28s%29+=++%289%2F%28s%2A%28s%5E2-3s%29%29%29+ ALIGN=MIDDLE  ALT=\"+Y%28s%29+=++%289%2F%28s%2A%28s%5E2-3s%29%29%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "   <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=++Y%28s%29+=++%289%2F%28s%5E2%2A%28s-3%29%29%29+ ALIGN=MIDDLE  ALT=\"++Y%28s%29+=++%289%2F%28s%5E2%2A%28s-3%29%29%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > <br>\r<BR>\n" );
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document.write( "Use partial fraction expansion to get the right hand side into a form in which the inverse Laplace can be taken:<BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=++%289%2F%28s%5E2%2A%28s-3%29%29%29+=+A%2Fs+%2B+B%2Fs%5E2+%2B+C%2F%28s-3%29+ ALIGN=MIDDLE  ALT=\"++%289%2F%28s%5E2%2A%28s-3%29%29%29+=+A%2Fs+%2B+B%2Fs%5E2+%2B+C%2F%28s-3%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >   (1)\r<BR>\n" );
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document.write( "Multiply both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+s%5E2%28s-3%29+ ALIGN=MIDDLE  ALT=\"+s%5E2%28s-3%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=++9+=++A%2A%28s%2A%28s-3%29%29+%2B+B%2A%28s-3%29+%2B+C%2As%5E2+ ALIGN=MIDDLE  ALT=\"++9+=++A%2A%28s%2A%28s-3%29%29+%2B+B%2A%28s-3%29+%2B+C%2As%5E2+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=++9+=++%28As%5E2+-+3As%29+%2B+%28Bs+-+3B%29+%2B+Cs%5E2+ ALIGN=MIDDLE  ALT=\"++9+=++%28As%5E2+-+3As%29+%2B+%28Bs+-+3B%29+%2B+Cs%5E2+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "From this you get three equations in three unknowns:<BR>\n" );
document.write( "           A+C = 0     (from the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s%5E2 ALIGN=MIDDLE  ALT=\"s%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > terms)<BR>\n" );
document.write( "           -3A+B = 0    (from the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s%5E1 ALIGN=MIDDLE  ALT=\"s%5E1\"   DISABLED_style= \"max-width:90%; height:auto;\" > terms)<BR>\n" );
document.write( "           -3B = 9        (from the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s%5E0 ALIGN=MIDDLE  ALT=\"s%5E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > terms)\r<BR>\n" );
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document.write( "—>  B = -3  —> A= -1   —>  C = 1  <br><BR>\n" );
document.write( "            <BR>\n" );
document.write( "So we can write (1) as:\r<BR>\n" );
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document.write( "   <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=++Y%28s%29+=+-1%2Fs+%2B+-3%2Fs%5E2+%2B+1%2F%28s-3%29++ ALIGN=MIDDLE  ALT=\"++Y%28s%29+=+-1%2Fs+%2B+-3%2Fs%5E2+%2B+1%2F%28s-3%29++\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Taking the inverse Laplace gives:<BR>\n" );
document.write( "   <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+highlight%28++y%28x%29++=++-1+-3x+%2B+e%5E%283x%29+%29+ ALIGN=MIDDLE  ALT=\"+highlight%28++y%28x%29++=++-1+-3x+%2B+e%5E%283x%29+%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "——\r<BR>\n" );
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document.write( "Check:  <BR>\n" );
document.write( "Initial conditions:<BR>\n" );
document.write( "y(0) = -1-3*0+e^(0) = -1 + 1 = 0 (ok)<BR>\n" );
document.write( "y'(x) = -3 + 3e^(3x) and y'(0) = -3 + 3e^(0) = -3+3 = 0  (also ok)\r<BR>\n" );
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document.write( "Entire equation:<BR>\n" );
document.write( "  y''(x) = 9e^(3x)\r<BR>\n" );
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document.write( "   y'' - 3y' = 9e^(3x) - 3(-3+3e^(3x)) = 9e^(3x) + 9 -9e^(3x) = 9  (ok)\r<BR>\n" );
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document.write( "———————————\r<BR>\n" );
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document.write( "My answer is the general solution.   A particular solution is often guessed at the start, and then combined with the homogeneous solution (i.e. particular solution would be a function y(x) that satisfies y''-3y' = 9 while the homogenous (\"complementary\") solution would satisfy y''-3y' = 0 and you  add the two solutions together to get the general solution.  I don't know how to guess a proper particular solution for this problem.  One could guess y(x) = Ae^(kx) + Bxe^(mx) + C,  I suppose, but I wouldn't know to guess that without seeing the general solution first.  \r<BR>\n" );
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document.write( " \r<BR>\n" );
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document.write( "  <BR>\n" );
document.write( " </SPAN>\n" );
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