document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1115049>Question 1115049</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<BR>\n" );
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document.write( "given the initial conditions:<BR>\n" );
document.write( "y(0) = 0,     y'(0) = 0<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 #730041 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=730041\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> this is a second-order linear differential equation which is nonhomogeneous <BR>\n" );
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document.write( "Note the general form is<BR>\n" );
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document.write( "1) a(d^2 y/dx^2) + b(dy/dx) + c(x)y = G(x) where a, b, c, are constants and G is a continuous functions not equal to 0 for some x<BR>\n" );
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document.write( "The complementary form is<BR>\n" );
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document.write( "2) a(d^2 y/dx^2) + b(dy/dx) + c(x)y = G(x) where a, b, c, are constants and G is a continuous functions equal to 0 for some x<BR>\n" );
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document.write( "The general solution is given by the formula<BR>\n" );
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document.write( "y(x) = y(p) (x) + y(c) (x), where y(p) is a particular solution to equation 1 and y(c) is a general solution to equation 2<BR>\n" );
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document.write( "we are given the problem<BR>\n" );
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document.write( "(d^2 y/d x^2) - 3(dy/dx) = 9<BR>\n" );
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document.write( "solve the equation 2 form<BR>\n" );
document.write( "r^2 -3r = 0<BR>\n" );
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document.write( "r(r-3) = 0<BR>\n" );
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document.write( "r=0 and r=3<BR>\n" );
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document.write( "So the solution to equation 2 is<BR>\n" );
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document.write( "y(c) = c(1)e^(0 * x) + c(2)e^-3x = c(1) +c(2)e^(-3x)<BR>\n" );
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document.write( "since G(x) = 9 is a linear equation, we are looking for a particular solution of the form<BR>\n" );
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document.write( "y(p) (x) = Bx +C<BR>\n" );
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document.write( "y'(p) = B and y''(p) = 0<BR>\n" );
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document.write( "now substitute into the given equation<BR>\n" );
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document.write( "0 -3B = 9<BR>\n" );
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document.write( "polynomials are equal when their coefficients are equal<BR>\n" );
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document.write( "B = -3<BR>\n" );
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document.write( "the general solution is<BR>\n" );
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document.write( "y(x) = c(1) + c(2)e^(-3x) + 9<BR>\n" );
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document.write( "y(0)=0, then<BR>\n" );
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document.write( "c(1) +c(2) +9 = 0<BR>\n" );
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document.write( "y' = -3c(2)e^(-3x)<BR>\n" );
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document.write( "y'(0) = 0<BR>\n" );
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document.write( "0 = -3c(2)<BR>\n" );
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document.write( "c(2) = 0 and c(1) = -9<BR>\n" );
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document.write( "the solution to the initial value problem is<BR>\n" );
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document.write( "y(x) = -9 +0 +9 = 0<BR>\n" );
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