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<title> 常微分方程式 </title>
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<font COLOR="#0000FF">
<h1>11.2.2 常微分方程式</h1>
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<p>一阶常微分方程式 <font FACE="Times New Roman">(first-order ordinary
differential equation, ODE) </font>可写为 </p>
<p><img SRC="../img11/img00004.gif" tppabs="http://webclass.ncu.edu.tw/~junwu/img11/img00004.gif" WIDTH="130" HEIGHT="44"> </p>
<p>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</font>'</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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<p>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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<p>假设有以下三个一阶常微分方程式和其初始条件 </p>
<p><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> </p>
<p><i>y'</i>=2<sup>.</sup>x<sup>.</sup>cos(y)<sup>2</sup>, <i>y</i>(0)=0.25<img SRC="../img11/img00005.gif" tppabs="http://webclass.ncu.edu.tw/~junwu/img11/img00005.gif" WIDTH="16" HEIGHT="16">
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<p><i><font FACE="Times New Roman">y</font>'</i><font FACE="Times New Roman">=3y+exp(2x),
y(0)=3</font> </p>
<p>对应上述常微分方程式的符号运算式为： </p>
<p><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> </p>
<p><font COLOR="#FF0000" FACE="Times New Roman">ans=</font> </p>
<p><font COLOR="#FF0000" FACE="Times New Roman">x^3-7.500000000000000</font> </p>
<p><font COLOR="#FF0000" FACE="Times New Roman">>>ezplot(soln_1,[2,4]) % </font>看看这个函数的长相
</p>
<p><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> </p>
<p><font COLOR="#FF0000" FACE="Times New Roman">ans=</font> </p>
<p><font COLOR="#FF0000" FACE="Times New Roman">atan(x^2+1)</font> </p>
<p><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> </p>
<p><font COLOR="#FF0000" FACE="Times New Roman">ans=</font> </p>
<p><font COLOR="#FF0000" FACE="Times New Roman">-exp(2*x)+4*exp(3*x)</font></p>
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