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<TITLE> 常微分方程式 </TITLE>
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<H1>11.2.2 常微分方程式</H1>
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一阶常微分方程式<FONT FACE="Times New Roman"> (first-order ordinary
differential equation, ODE) </FONT>可写为
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其中<I><FONT FACE="Times New Roman">x</FONT></I>为独立变数，而<I><FONT FACE="Times New Roman">y</FONT></I>是<I><FONT FACE="Times New Roman">x</FONT></I>的函数。上述的一阶常微分方程式的解是<FONT FACE="Times New Roman">
<I>y</I>=<I>f</I>(<I>x</I>,<I>y</I>)</FONT>可以满足<I><FONT FACE="Times New Roman">y</FONT>'</I><FONT FACE="Times New Roman">=<I>f</I></FONT><I>'</I><FONT FACE="Times New Roman">=g(<I>x</I>,<I>y</I>)</FONT>。关于常微分
方程式的解法已再第十章说明过，它还需要初始条件才能得到为一的解。
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MATLAB解常微分方程式的语法是<FONT COLOR=#FF0000 FACE="Times New Roman">dsolve(</FONT><FONT COLOR=#FF0000>'</FONT><FONT COLOR=#FF0000 FACE="Times New Roman">equation</FONT><FONT COLOR=#FF0000>'</FONT><FONT COLOR=#FF0000 FACE="Times New Roman">,</FONT><FONT COLOR=#FF0000>'</FONT><FONT COLOR=#FF0000 FACE="Times New Roman">condition</FONT><FONT COLOR=#FF0000>'</FONT><FONT COLOR=#FF0000 FACE="Times New Roman">)</FONT>，其中<FONT COLOR=#FF0000 FACE="Times New Roman">equation</FONT>代表常微分方程式即<I><FONT FACE="Times New Roman">y</FONT>'</I><FONT FACE="Times New Roman">=g(<I>x</I>,<I>y</I>)</FONT>，且须
以<FONT COLOR=#FF0000 FACE="Times New Roman">Dy</FONT>代表一阶微分项<I><FONT FACE="Times New Roman">y</FONT>'</I> <FONT COLOR=#FF0000 FACE="Times New Roman">D2y</FONT>代表二阶微分项<I><FONT FACE="Times New Roman">y</FONT>''</I> ，<FONT COLOR=#FF0000 FACE="Times New Roman">condition</FONT>则为初始条件。
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假设有以下三个一阶常微分方程式和其初始条件
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<I><FONT FACE="Times New Roman">y</FONT>'</I><FONT FACE="Times New Roman">=3<I>x</I><SUP>2</SUP>,
<I>y</I>(2)=0.5</FONT>
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<I>y'</I>=2<SUP>.</SUP>x<SUP>.</SUP>cos(y)<SUP>2</SUP>, <I>y</I>(0)=0.25<IMG SRC="img00005-8.gif" tppabs="http://166.111.167.223/computer/cai/matlabjc/img11/img00005.gif">
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<I><FONT FACE="Times New Roman">y</FONT>'</I><FONT FACE="Times New Roman">=3y+exp(2x),
y(0)=3</FONT>
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对应上述常微分方程式的符号运算式为：
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<FONT COLOR=#FF0000 FACE="Times New Roman">>>soln_1 = dsolve(</FONT><FONT COLOR=#FF0000>'</FONT><FONT COLOR=#FF0000 FACE="Times New Roman">Dy
= 3*x^2</FONT><FONT COLOR=#FF0000>'</FONT><FONT COLOR=#FF0000 FACE="Times New Roman">,</FONT><FONT COLOR=#FF0000>'</FONT><FONT COLOR=#FF0000 FACE="Times New Roman">y(2)=0.5</FONT><FONT COLOR=#FF0000>'</FONT><FONT COLOR=#FF0000 FACE="Times New Roman">)</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">ans=</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">x^3-7.500000000000000</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">>>ezplot(soln_1,[2,4])
% </FONT>看看这个函数的长相
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<FONT COLOR=#FF0000 FACE="Times New Roman">>>soln_2 = dsolve(</FONT><FONT COLOR=#FF0000>'</FONT><FONT COLOR=#FF0000 FACE="Times New Roman">Dy
= 2*x*cos(y)^2</FONT><FONT COLOR=#FF0000>'</FONT><FONT COLOR=#FF0000 FACE="Times New Roman">,</FONT><FONT COLOR=#FF0000>'</FONT><FONT COLOR=#FF0000 FACE="Times New Roman">y(0)
= pi/4</FONT><FONT COLOR=#FF0000>'</FONT><FONT COLOR=#FF0000 FACE="Times New Roman">)</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">ans=</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">atan(x^2+1)</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">>>soln_3 = dsolve(</FONT><FONT COLOR=#FF0000>'</FONT><FONT COLOR=#FF0000 FACE="Times New Roman">Dy
= 3*y + exp(2*x)</FONT><FONT COLOR=#FF0000>'</FONT><FONT COLOR=#FF0000 FACE="Times New Roman">,</FONT><FONT COLOR=#FF0000>'</FONT><FONT COLOR=#FF0000 FACE="Times New Roman">
y(0) = 3</FONT><FONT COLOR=#FF0000>'</FONT><FONT COLOR=#FF0000 FACE="Times New Roman">)</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">ans=</FONT>
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<FONT COLOR=#FF0000 FACE="Times New Roman">-exp(2*x)+4*exp(3*x)</FONT><HR>
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