document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=212136>Question 212136</A>: <I><SPAN STYLE=\"word-break: break-all;\"> i need a solution to the ordinary differential equation <BR>\n" );
document.write( "x\"+2x'+2x=0 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #160217 by </B><A HREF=/tutors/aboutme.mpl?userid=drj><B>drj(1380)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=drj><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=drj><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=160217\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I need a solution to the ordinary differential equation <BR>\n" );
document.write( "x\"+2x'+2x=0\r<BR>\n" );
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document.write( "Step 1.  Assume the solution is \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=e%5Emx ALIGN=MIDDLE  ALT=\"x=e%5Emx\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "If this is true, then we need to solve for m.\r<BR>\n" );
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document.write( "Step 2. Take the derivatives of x:\r<BR>\n" );
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document.write( "x\" =  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5E2e%5Emx ALIGN=MIDDLE  ALT=\"m%5E2e%5Emx\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "x' = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=me%5Emx+ ALIGN=MIDDLE  ALT=\"me%5Emx+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Step 3.  Substitute above derivatives and x into the given equation\r<BR>\n" );
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document.write( "x\"+2x'+2x= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0=m%5E2e%5Emx%2B2me%5Emx%2B2e%5Emx ALIGN=MIDDLE  ALT=\"0=m%5E2e%5Emx%2B2me%5Emx%2B2e%5Emx\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Step 4.  Factor out\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5Emx ALIGN=MIDDLE  ALT=\"e%5Emx\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "to get\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28e%5Emx%29%28m%5E2%2B2m%2B2%29=0 ALIGN=MIDDLE  ALT=\"%28e%5Emx%29%28m%5E2%2B2m%2B2%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Step 5.  The only way to get 0 is that the quadratic expression is zero.  That is,\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5E2%2B2m%2B2=0 ALIGN=MIDDLE  ALT=\"m%5E2%2B2m%2B2=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now, we can solve for m using the quadratic formula below.\r<BR>\n" );
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document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+ ALIGN=MIDDLE  ALT=\"m+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "where \r<BR>\n" );
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document.write( "a=1, b=2 and c=2\r<BR>\n" );
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document.write( "Step 6.  See standard procedure of solving quadratic equation below:\r<BR>\n" );
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document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=quadratic.solver>SOLVE quadratic equation with variable</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT>Quadratic equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=am%5E2%2Bbm%2Bc=0 ALIGN=MIDDLE  ALT=\"am%5E2%2Bbm%2Bc=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (in our case <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1m%5E2%2B2m%2B2+=+0 ALIGN=MIDDLE  ALT=\"1m%5E2%2B2m%2B2+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >) has the following solutons:<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"m%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  For these solutions to exist, the <U>discriminant</U> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" > should not be a negative number.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  First, we need to compute the discriminant <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" >: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac=%282%29%5E2-4%2A1%2A2=-4 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%282%29%5E2-4%2A1%2A2=-4\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
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document.write( "  The discriminant -4 is less than zero. That means that there are no solutions among real numbers.<BR><BR>\n" );
document.write( "  If you are a student of advanced school algebra and are aware about <u>imaginary numbers<u>, read on.<BR><BR>\n" );
document.write( "  <BR>\n" );
document.write( "      In the field of imaginary numbers, the square root of -4 is + or - <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28+4%29+=+2 ALIGN=MIDDLE  ALT=\"sqrt%28+4%29+=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. <BR>\n" );
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document.write( "      The solution is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5B12%5D+=+%28-2%2B-+i%2Asqrt%28+-4+%29%29%2F2%5C1+=++%28-2%2B-+i%2A2%29%2F2%5C1+ ALIGN=MIDDLE  ALT=\"m%5B12%5D+=+%28-2%2B-+i%2Asqrt%28+-4+%29%29%2F2%5C1+=++%28-2%2B-+i%2A2%29%2F2%5C1+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Here's your graph:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B2%2Ax%2B2+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B2%2Ax%2B2+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
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document.write( "Step 7.  Note the roots are complex.  So your solution will consists of complex exponentials.  Also, please ignore the graph since it's only applicable for real roots.\r<BR>\n" );
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document.write( "Using the above roots m1 and m2, the solution is\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=c1e%5Em1%2Bc2e%5Em2 ALIGN=MIDDLE  ALT=\"y=c1e%5Em1%2Bc2e%5Em2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "where the c1 and c2 are arbitrary constants.   We need initial values to get values of c1 and c2.\r<BR>\n" );
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document.write( "I believe the above problem is above the skill level of algebra.  However,<BR>\n" );
document.write( "for Step-By-Step Videos on Differential Equations, Circuit Analysis and Design,please visit http://wwww.FreedomUniversity.TV\r<BR>\n" );
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