Method of Solution

<BODY LANG="EN"> <HR><center> <table border=1> <TR> <td><A HREF="node8.shtml"><img height=36 width=36 alt="PREVIOUS" src="/UCESIcons/tools/previous.gif"></A></td> <td><A HREF="cone.shtml"><img height=36 width=36 alt="UP" src="/UCESIcons/tools/up.gif"></A></td> <td><A HREF="node10.shtml"><img height=36 width=36 alt="NEXT" src="/UCESIcons/tools/next.gif"></A></td> <td><a href="contents-node9.html"><img height=36 width=36 alt="CONTENTS" src="/UCESIcons/tools/toc.gif"></a></td> <td><a href="reset.hamlet"><img height=36 width=36 alt="RESET" src="/UCESIcons/tools/stop.gif"></a></td> <td><a href="copyright.html"><img height=36 width=36 alt="COPYRIGHT" src="/UCESIcons/tools/copyright.gif"></a></td> <td><a href="mailto:uces_info@krellinst.org"><img height=36 width=36 alt="FEEDBACK" src="/UCESIcons/tools/feedback.gif"></a></td> <td><a href="http://www.cs.utah.edu/%7Ehamlet/icons.html"><img height=36 width=36 alt="ABOUT" src="/UCESIcons/tools/about.gif"></a></td> </TR> </table> </center> <HR> <!DOCTYPE HTML PUBLIC "-//IETF//DTD HTML 2.2//EN">  <HTML> <HEAD> <TITLE>Method of Solution</TITLE> <BASE TARGET="_parent"> <META NAME="description" CONTENT="Method of Solution"> <META NAME="keywords" CONTENT="cone"> <META NAME="resource-type" CONTENT="document"> <META NAME="distribution" CONTENT="global"> <LINK REL=STYLESHEET HREF="cone.css"> </HEAD> <BODY LANG="EN"> <H1>Method of Solution</H1> <P> The two equations we arrived at above are both ordinary differential equations. In principle, we should have enough tools at our disposal to solve them by hand, given enough time to eliminate all the errors. Before we decide how to attack them <EM>mathematically</EM>, however, let's rearrange them a bit. <P> For the circular cylinder, we can easily separate the variables, putting all functions of <I>y</I> on one side and all the constants and <I>t</I> on the other side: <P> <IMG WIDTH=500 HEIGHT=43 ALIGN=BOTTOM ALT="[Eq. 17]" SRC="img46.gif"> <P> Rearranging the equation for the conical tank requires a bit more algebra: <P> <IMG WIDTH=533 HEIGHT=65 ALIGN=BOTTOM ALT="[Eq. 18]" SRC="img47.gif"> <P> As noted above, <I>r</I> is the minor radius of the cone section, <IMG WIDTH=17 HEIGHT=22 ALIGN=MIDDLE ALT="R_0" SRC="img36.gif"> is the major radius, and <IMG WIDTH=12 HEIGHT=14 ALIGN=MIDDLE ALT="r_e" SRC="img12.gif"> is the radius of the drain. <P> Both of these equations appear to be soluble analytically, although (13) may be quite tedious. In either case, it may not be possible to find an explicit expression for <nobr><I>y</I>(<I>t</I>).</nobr> That is, the solution may be in the form of a transcendental or some other function which does not allow us to isolate the solution on the left and the independent variables and constants on the right. In that case, it may make sense to evaluate <nobr><I>t</I>(<I>y</I>),</nobr> if that is more easily plotted. <P> </BODY> </HTML> <HR><center> <table border=1> <TR> <td><A HREF="node8.shtml"><img height=36 width=36 alt="PREVIOUS" src="/UCESIcons/tools/previous.gif"></A></td> <td><A HREF="cone.shtml"><img height=36 width=36 alt="UP" src="/UCESIcons/tools/up.gif"></A></td> <td><A HREF="node10.shtml"><img height=36 width=36 alt="NEXT" src="/UCESIcons/tools/next.gif"></A></td> <td><a href="contents-node9.html"><img height=36 width=36 alt="CONTENTS" src="/UCESIcons/tools/toc.gif"></a></td> <td><a href="reset.hamlet"><img height=36 width=36 alt="RESET" src="/UCESIcons/tools/stop.gif"></a></td> <td><a href="copyright.html"><img height=36 width=36 alt="COPYRIGHT" src="/UCESIcons/tools/copyright.gif"></a></td> <td><a href="mailto:uces_info@krellinst.org"><img height=36 width=36 alt="FEEDBACK" src="/UCESIcons/tools/feedback.gif"></a></td> <td><a href="http://www.cs.utah.edu/%7Ehamlet/icons.html"><img height=36 width=36 alt="ABOUT" src="/UCESIcons/tools/about.gif"></a></td> </TR> </table> </center> <HR> <P> <ADDRESS> <I>Boyd Gatlin<BR> Associate Professor of Aerospace Engineering<BR> Mississippi State University</I>  </ADDRESS> </BODY>