Semantics And Examples

<BODY LANG="EN" > <HR><center> <table border=1> <td><A HREF="diffeq.html"><img height=36 width=36 alt="PREVIOUS" src="/UCESIcons/tools/previous.gif"></A></td> <td><A HREF="diffeq.html"><img height=36 width=36 alt="UP" src="/UCESIcons/tools/up.gif"></A></td> <td><A HREF="node2.html"><img height=36 width=36 alt="NEXT" src="/UCESIcons/tools/next.gif"></A></td> <td><a href="contents-node1.html"><img height=36 width=36 alt="CONTENTS" src="/UCESIcons/tools/toc.gif"></a></td> </table> </center> <HR> <H1>Semantics And Examples</H1> Let us introduce (or remember) some terminology that will be used in this chapter. <P> We start with the general form for a <EM>first order</EM> differential equation (order referring to the power of the derivative) <P> <A NAME="diffE"><IMG WIDTH=500 HEIGHT=75 ALIGN=BOTTOM ALT="[1st order DE]" SRC="img6.gif"></A> <P> where the derivative function <I>f</I>(<I>t</I>,<I>y</I>) is arbitrary. Likewise, a <EM>second order</EM> differential equation has the form <P> <A NAME="diffE2"><IMG WIDTH=500 HEIGHT=82 ALIGN=BOTTOM ALT="[2nd order DE]" SRC="img7.gif"></A> <P> This should be familiar from Newton's law (<A HREF="diffeq.html#newton">1</A>). In the differential equations (<A HREF="node1.html#diffE">4</A>) and (<A HREF="node1.html#diffE2">6</A>), the variable <I>t</I> is <EM>independent</EM> while <I>y</I> is <EM>dependent</EM>, that is, we imagine being able to vary or measure any value of <I>t</I> as desired, with the corresponding value of <I>y</I>(<I>t</I>) depending on our choice for <I>t</I>. (In many applications <I>t</I> will be time -- but it really is an arbitrary variable). <P> The differential equations (<A HREF="diffeq.html#newton">1</A>) and (<A HREF="node1.html#diffE">4</A>) are all <EM>ordinary</EM>, whereas the Schrödinger equation (<A HREF="diffeq.html#SE">2</A>) is a <EM>partial</EM> differential equation. In this chapter we discuss ordinary differential equations. Since partial differential equations are often solved by reducing them to ordinary equations, the current discussions will still be useful for partial differential equations. In a later chapter we examine some techniques especially applicable to partial equations. <P> One of the nice things about solving differential equations with computers is that we are not limited to the <EM>linear equations</EM> accessible to analytic techniques. A <EM>linear</EM> equation contains only the first power of <I>y</I> or <I>dy</I>/<I>dt</I> in them, while a <EM>nonlinear</EM> equation is not so restricted. For example <P> <A NAME="linear"><IMG WIDTH=500 HEIGHT=38 ALIGN=BOTTOM ALT="[linear eqn]" SRC="img8a.gif"></A><BR> <A NAME="nonlinear"><IMG WIDTH=500 HEIGHT=38 ALIGN=BOTTOM ALT="[nonlinear eqn]" SRC="img8b.gif"></A> <P> One of the nice properties of linear equations is that the <A HREF="node15.html#supp"><I>principle of linear superposition</I></A> holds, that is, if <I>a</I>(<I>t</I>) and <I>b</I>(<I>t</I>) are solutions of (<A HREF="node1.html#linear">8</A>), then <IMG WIDTH=96 HEIGHT=15 ALIGN=ABSMIDDLE ALT="alpha*a(t) + beta*b(t)" SRC="img9.gif"> is also a solution for arbitrary values of the constants <IMG WIDTH=7 HEIGHT=7 ALIGN=BOTTOM ALT="alpha" SRC="img10.gif"> and <IMG WIDTH=9 HEIGHT=15 ALIGN=ABSMIDDLE ALT="beta" SRC="img11.gif">. This is <EM>not</EM> true for nonlinear equations, a fact we invite the reader to verify with the following nonlinear equation and its analytic solution: <P> <IMG WIDTH=500 HEIGHT=71 ALIGN=BOTTOM ALT="eqnarray94" SRC="img12.gif"> <P> For a given derivative function <I>f</I> there are entire classes of solutions to (<A HREF="node1.html#diffE">4</A>). The specific solution for a specific problem is required to satisfy <EM>initial conditions</EM> specific to that problem. For our first-order equation, the initial condition is usually the specification of the value of the <I>y</I>(<I>t</I>) at some time, for example <IMG WIDTH=94 HEIGHT=15 ALIGN=ABSMIDDLE ALT="y(t=0)=y0" SRC="img13.gif">. Higher order equations need more than one initial condition, and often these will be values of the <I>y</I>(<I>t</I>) and <I>y</I>'(<I>t</I>) at one ``time''. <P> If not all initial conditions are at the same ``time'' <I>t</I>, you will need modify the strategies discussed here to obtain solutions. In particular, there is a class of solutions to differential equations known as <EM>boundary value problems</EM> in which the solution is constrained to have some property at more than one point in space (and possibly an entire region for a multidimensional problem). Problems of this sort are usually more demanding and require a <EM>search</EM> to find the particular values of parameters (called <EM>eigenvalues</EM>) for which solutions exist. We discuss them in a later chapter. In this chapter we consider the oscillations of a particle when special forces are applied. <P> <HR><center> <table border=1> <td><A HREF="diffeq.html"><img height=36 width=36 alt="PREVIOUS" src="/UCESIcons/tools/previous.gif"></A></td> <td><A HREF="diffeq.html"><img height=36 width=36 alt="UP" src="/UCESIcons/tools/up.gif"></A></td> <td><A HREF="node2.html"><img height=36 width=36 alt="NEXT" src="/UCESIcons/tools/next.gif"></A></td> <td><a href="contents-node1.html"><img height=36 width=36 alt="CONTENTS" src="/UCESIcons/tools/toc.gif"></a></td> </table> </center> <HR> </BODY>