document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=607237>Question 607237</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the Wronskian of 1, cos2x, sin²x at x=PI/2. Are these functions dependent? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #382529 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=382529\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "General 3X3 Wronskian:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \left|f\ \ \ \ \,g\ \ \ \ \,h\cr f'\ \ \ g'\ \ \ h'\cr f^{[2]}\  g^{[2]}\ h^{[2]}\right|\">\r<BR>\n" );
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document.write( "For your functions:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \left|1\ \ \ \ \,\cos(2x)\ \ \ \ \ \sin^2(x)\cr 0\ \ \ -2\sin(2x)\ \ \ 2\cos(x)\sin(x)\cr 0\ \ \ -4\cos(2x)\ \ \ 2\cos^2(x)-2\sin^2(x)\right|\">\r<BR>\n" );
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document.write( "At <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x\ =\ \frac{\pi}{2}\">:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \left|1\ -1\ \ \ 1\cr 0\ \ \,0\ \ \ 0\cr 0\ \ \,4\ -2\right|\ =\ 0\">\r<BR>\n" );
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document.write( "Verification of the value of the discriminant left as an exercise for the student.  The Wronskian is zero, so the system <i><b>could be</b></i> dependent, but not necessasarily.\r<BR>\n" );
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document.write( "If they are linearly dependent, then you should be able to find constants <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE c_1,\ c_2,\ c_3\">, two of which must be non-zero such that:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ c_1\ +\ c_2\cos(2x)\ +\ c_3\sin^2(x)\ =\ 0\">\r<BR>\n" );
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document.write( "for all <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\">.  Intuitively speaking, I'd say they are independent.\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi}\ +\ 1\ =\ 0\"><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
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