document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1184638>Question 1184638</A>: <I><SPAN STYLE=\"word-break: break-all;\"> For this problem you may want to recall that if y=sinh(x), then dy/dx=cosh(x), and d^(2)y/dx^(2)=sinh(x).\r<BR>\n" );
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document.write( "It can be shown that y1=sinh(3x), y2=cosh(3x) and y3=e^(3x) are solutions to the differential equation xD^(3)y−8D^(2)y−9xDy+72y=0 on (0,∞).\r<BR>\n" );
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document.write( "a)<BR>\n" );
document.write( "What does the Wronskian of y1,y2,y3 equal on (0,∞)?<BR>\n" );
document.write( "W(y1,y2,y3)=______________________ on (0,∞).\r<BR>\n" );
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document.write( "b)<BR>\n" );
document.write( "Is {y1,y2,y3} a fundamental set for xD^(3)y−8D^(2)y−9xDy+72y=0 on (0,∞)?<BR>\n" );
document.write( "Yes or No </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #815296 by </B><A HREF=/tutors/aboutme.mpl?userid=robertb><B>robertb(5830)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=robertb><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=robertb><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=815296\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> \r<BR>\n" );
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document.write( "a)  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=W%28y%5B1%5D%2Cy%5B2%5D%2Cy%5B3%5D%29+=+abs%28matrix%283%2C3%2C+sinh%283x%29%2C+cosh%283x%29%2C+e%5E%283x%29%2C+3cosh%283x%29%2C+3sinh%283x%29%2C+3e%5E%283x%29%2C+9sinh%283x%29%2C+9cosh%283x%29%2C+9e%5E%283x%29%29%29+=+%0D%0A%0D%0A%0D%0A27e%5E%283x%29abs%28matrix%283%2C3%2C+sinh%283x%29%2C+cosh%283x%29%2C+1%2C+cosh%283x%29%2C+sinh%283x%29%2C+1%2C+sinh%283x%29%2C+cosh%283x%29%2C+1%29%29+=+0 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "***This is expected since the set { <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B1%5D ALIGN=MIDDLE  ALT=\"y%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B2%5D ALIGN=MIDDLE  ALT=\"y%5B2%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+y%5B3%5D ALIGN=MIDDLE  ALT=\"+y%5B3%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >} is linearly dependent, because <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B1%5D+%2B+y%5B2%5D+=+sinh%283x%29+%2B+cosh%283x%29+=+e%5E%283x%29=y%5B3%5D ALIGN=MIDDLE  ALT=\"y%5B1%5D+%2B+y%5B2%5D+=+sinh%283x%29+%2B+cosh%283x%29+=+e%5E%283x%29=y%5B3%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "b)  No, { <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B1%5D ALIGN=MIDDLE  ALT=\"y%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B2%5D ALIGN=MIDDLE  ALT=\"y%5B2%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+y%5B3%5D ALIGN=MIDDLE  ALT=\"+y%5B3%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >} is not a fundamental set for the D.E. (i.e., it cannot be a basis for the solutions of the D.E.). </SPAN>\n" );
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