document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1030555>Question 1030555</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The differential equation dy/dx = y-3/x-3.\r<BR>\n" );
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document.write( "produces a slope field with horizontal tangents at y = 3<BR>\n" );
document.write( "produces a slope field with vertical tangents at y = 3<BR>\n" );
document.write( "produces a slope field with rows of parallel segments<BR>\n" );
document.write( " I only<BR>\n" );
document.write( " II only<BR>\n" );
document.write( " I and II<BR>\n" );
document.write( " III only </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #645789 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=645789\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I only\r<BR>\n" );
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document.write( "II cannot be true.  The DE is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dy%2Fdx+=+%28y-3%29%2F%28x-3%29 ALIGN=MIDDLE  ALT=\"dy%2Fdx+=+%28y-3%29%2F%28x-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Letting y = 3 will give <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dy%2Fdx+=+0 ALIGN=MIDDLE  ALT=\"dy%2Fdx+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3C%3E3 ALIGN=MIDDLE  ALT=\"x%3C%3E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >), and hence corresponds to horizontal, not vertical tangents.\r<BR>\n" );
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document.write( "III cannot be true.  Solving for the general solution of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dy%2Fdx+=+%28y-3%29%2F%28x-3%29 ALIGN=MIDDLE  ALT=\"dy%2Fdx+=+%28y-3%29%2F%28x-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >,\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dy%2F%28y-3%29+=+dx%2F%28x-3%29 ALIGN=MIDDLE  ALT=\"dy%2F%28y-3%29+=+dx%2F%28x-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ==> ln(y-3) = ln(x-3) + lnk for arbitrary k.\r<BR>\n" );
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document.write( "==> ln(y-3) = ln(k(x-3)) ==>  y-3 = k(x-3).\r<BR>\n" );
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document.write( "The general solution yields a family of straight lines that pass through the point (3,3) with the values of the slope varying through all the real numbers.  In other words, drawing an arbitrary horizontal line on the cartesian plane will intersect this family at a row of infinite number of points where the (tangential) segments are not parallel, but in which the slopes vary from negative to positive infinity. </SPAN>\n" );
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