document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=199856>Question 199856</A>: <I><SPAN STYLE=\"word-break: break-all;\">  (s\") + 6s' - 9s = t^2 <BR>\n" );
document.write( " where s\" is like<BR>\n" );
document.write( " d^2x/dt^2<BR>\n" );
document.write( " (s\") + 6s' - 9s = t^2 <BR>\n" );
document.write( " where s\" is like<BR>\n" );
document.write( " d^2x/dt^2 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #150236 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=150236\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> This is a 2nd order linear differential equation. So here are the steps to finding the general solution:\r<BR>\n" );
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document.write( "Step 1) Find the complementary solution <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s%5Bc%5D ALIGN=MIDDLE  ALT=\"s%5Bc%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Solve the characteristic equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5E2%2B6r-9=0 ALIGN=MIDDLE  ALT=\"r%5E2%2B6r-9=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=-3%2B3%2Asqrt%282%29 ALIGN=MIDDLE  ALT=\"r=-3%2B3%2Asqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=-3-3%2Asqrt%282%29 ALIGN=MIDDLE  ALT=\"r=-3-3%2Asqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Since we have two unique real roots, this means that the complementary solution is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s%5Bc%5D=c%5B1%5D%2Ae%5E%28%28-3%2B3%2Asqrt%282%29%29x%29%2Bc%5B2%5D%2Ae%5E%28%28-3-3%2Asqrt%282%29%29x%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Note: recall, if you have two real roots <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5B1%5D ALIGN=MIDDLE  ALT=\"r%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5B2%5D ALIGN=MIDDLE  ALT=\"r%5B2%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > of the characteristic equation, then the complementary solution is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s%5Bc%5D=c%5B1%5D%2Ae%5E%28r%5B1%5D%2Ax%29%2Bc%5B2%5D%2Ae%5E%28r%5B2%5D%2Ax%29 ALIGN=MIDDLE  ALT=\"s%5Bc%5D=c%5B1%5D%2Ae%5E%28r%5B1%5D%2Ax%29%2Bc%5B2%5D%2Ae%5E%28r%5B2%5D%2Ax%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Step 2) Find the operator that annihilates the right hand side <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%5E2 ALIGN=MIDDLE  ALT=\"t%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The operator <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D%5E3 ALIGN=MIDDLE  ALT=\"D%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > annihilates <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t ALIGN=MIDDLE  ALT=\"t\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%5E2 ALIGN=MIDDLE  ALT=\"t%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "We can rewrite the given differential equation as: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D%5E2%2B6D-9=t%5E2 ALIGN=MIDDLE  ALT=\"D%5E2%2B6D-9=t%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "When we apply the operator <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D%5E3 ALIGN=MIDDLE  ALT=\"D%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > to both sides, we get: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D%5E3%28D%5E2%2B6D-9%29=0 ALIGN=MIDDLE  ALT=\"D%5E3%28D%5E2%2B6D-9%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Because <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D%5E3 ALIGN=MIDDLE  ALT=\"D%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > annihilates <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t ALIGN=MIDDLE  ALT=\"t\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%5E2 ALIGN=MIDDLE  ALT=\"t%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we know that the particular solution <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s%5Bp%5D ALIGN=MIDDLE  ALT=\"s%5Bp%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > is\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s%5Bp%5D=A%2BBt%2BCt%5E2 ALIGN=MIDDLE  ALT=\"s%5Bp%5D=A%2BBt%2BCt%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Derive <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s%5Bp%5D ALIGN=MIDDLE  ALT=\"s%5Bp%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE s_{p}^\prime = B+2Ct\">\r<BR>\n" );
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document.write( "Derive again to get:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE s_{p}^{\prime \prime} = 2C\">\r<BR>\n" );
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document.write( "Now plug this information into the original differential equation to get\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-9Ct%5E2%2B%2812Ct-9Bt%29%2B%28-9A%2B6B%2B2C%29=t%5E2 ALIGN=MIDDLE  ALT=\"-9Ct%5E2%2B%2812Ct-9Bt%29%2B%28-9A%2B6B%2B2C%29=t%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So this means that \r<BR>\n" );
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document.write( "Solve the system above to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=-10%2F81 ALIGN=MIDDLE  ALT=\"A=-10%2F81\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B=-4%2F27 ALIGN=MIDDLE  ALT=\"B=-4%2F27\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C=-1%2F9 ALIGN=MIDDLE  ALT=\"C=-1%2F9\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the particular solution is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s%5Bp%5D=-10%2F81-%284%2F27%29x-%281%2F9%29x%5E2 ALIGN=MIDDLE  ALT=\"s%5Bp%5D=-10%2F81-%284%2F27%29x-%281%2F9%29x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "This means that the general solution is \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s=s%5Bc%5D%2Bs%5Bp%5D ALIGN=MIDDLE  ALT=\"s=s%5Bc%5D%2Bs%5Bp%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s%5Bc%5D ALIGN=MIDDLE  ALT=\"s%5Bc%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s%5Bp%5D ALIGN=MIDDLE  ALT=\"s%5Bp%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > (what we found earlier) to get:\r<BR>\n" );
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document.write( "So that is the final answer.\r<BR>\n" );
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document.write( "Note: there is another way to solve this problem, but it involves nasty integrals. Even though there's a lot going on here, the solution method is really straightforward.<BR>\n" );
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