📄 matlab2.htm
字号:
lang=EN-US><SPAN style="mso-spacerun: yes"> </SPAN></SPAN></P>
<P class=MsoNormal><SPAN lang=EN-US>>>S1 = '6*x^3-4*x^2+b*x-5';<SPAN
style="mso-spacerun: yes"> </SPAN></SPAN></P>
<P class=MsoNormal><SPAN lang=EN-US>>>S2 = 'sin(a)';<SPAN
style="mso-spacerun: yes"> </SPAN></SPAN></P>
<P class=MsoNormal><SPAN lang=EN-US>>>S3 = 'sqrt(x)';<SPAN
style="mso-spacerun: yes"> </SPAN></SPAN></P>
<P class=MsoNormal><SPAN lang=EN-US>>>int(S1)<SPAN
style="mso-spacerun: yes"> </SPAN></SPAN></P>
<P class=MsoNormal><SPAN lang=EN-US>ans= 3/2*x^4-4/3*x^3+1/2*b*x^2-5*x<SPAN
style="mso-spacerun: yes"> </SPAN></SPAN></P>
<P class=MsoNormal><SPAN lang=EN-US>>>int(S2)<SPAN
style="mso-spacerun: yes"> </SPAN></SPAN></P>
<P class=MsoNormal><SPAN lang=EN-US>ans= -cos(a)<SPAN
style="mso-spacerun: yes"> </SPAN></SPAN></P>
<P class=MsoNormal><SPAN lang=EN-US>>>int(S3)<SPAN
style="mso-spacerun: yes"> </SPAN></SPAN></P>
<P class=MsoNormal><SPAN lang=EN-US>ans= 2/3*x^(3/2)<SPAN
style="mso-spacerun: yes"> </SPAN></SPAN></P>
<P class=MsoNormal><SPAN lang=EN-US>>>int(S3,'a','b')<SPAN
style="mso-spacerun: yes"> </SPAN></SPAN></P>
<P class=MsoNormal><SPAN lang=EN-US>ans= 2/3*b^(3/2)- 2/3*a^(3/2)<SPAN
style="mso-spacerun: yes"> </SPAN></SPAN></P>
<P class=MsoNormal><SPAN lang=EN-US>>>int(S3,0.5,0.6)<SPAN
style="mso-spacerun: yes"> </SPAN></SPAN></P>
<P class=MsoNormal><SPAN lang=EN-US>ans= 2/25*15^(1/2)-1/6*2^(1/2)<SPAN
style="mso-spacerun: yes"> </SPAN></SPAN></P>
<P class=MsoNormal><SPAN lang=EN-US>>>numeric(int(S3,0.5,0.6)) %
</SPAN><SPAN
style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">使用</SPAN><SPAN
lang=EN-US>numeric</SPAN><SPAN
style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">函数可以计算积分的数值</SPAN><SPAN
lang=EN-US><SPAN style="mso-spacerun: yes"> </SPAN></SPAN></P>
<P class=MsoNormal><SPAN lang=EN-US>ans= 0.0741<SPAN
style="mso-spacerun: yes"> </SPAN></SPAN></P>
<P class=MsoNormal><B style="mso-bidi-font-weight: normal"><SPAN lang=EN-US
style="FONT-SIZE: 14pt; mso-bidi-font-size: 10.0pt">2</SPAN></B><B
style="mso-bidi-font-weight: normal"><SPAN
style="FONT-SIZE: 14pt; FONT-FAMILY: 宋体; mso-bidi-font-size: 10.0pt; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">．</SPAN></B><B
style="mso-bidi-font-weight: normal"><SPAN lang=EN-US
style="FONT-SIZE: 14pt; mso-bidi-font-size: 10.0pt">3</SPAN></B><B
style="mso-bidi-font-weight: normal"><SPAN
style="FONT-SIZE: 14pt; FONT-FAMILY: 宋体; mso-bidi-font-size: 10.0pt; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">求解常微分方程式</SPAN></B><B
style="mso-bidi-font-weight: normal"><SPAN lang=EN-US
style="FONT-SIZE: 14pt; mso-bidi-font-size: 10.0pt"><SPAN
style="mso-spacerun: yes"> </SPAN><O:P></O:P></SPAN></B></P>
<P class=MsoNormal><SPAN lang=EN-US><SPAN style="mso-spacerun: yes">
</SPAN>MATLAB</SPAN><SPAN
style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">解常微分方程式的语法是</SPAN><SPAN
lang=EN-US>dsolve('equation','condition')</SPAN><SPAN
style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">，其中</SPAN><SPAN
lang=EN-US>equation</SPAN><SPAN
style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">代表常微分方程式即</SPAN><SPAN
lang=EN-US>y'=g(x,y)</SPAN><SPAN
style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">，且须以</SPAN><SPAN
lang=EN-US>Dy</SPAN><SPAN
style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">代表一阶微分项</SPAN><SPAN
lang=EN-US>y'</SPAN><SPAN
style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'"> </SPAN><SPAN
lang=EN-US>D2y</SPAN><SPAN
style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">代表二阶微分项</SPAN><SPAN
lang=EN-US>y''</SPAN><SPAN
style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'"> ，</SPAN><SPAN
lang=EN-US><SPAN style="mso-spacerun: yes"> </SPAN></SPAN></P>
<P class=MsoNormal><SPAN lang=EN-US>condition</SPAN><SPAN
style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">则为初始条件。</SPAN><SPAN
lang=EN-US><SPAN style="mso-spacerun: yes">
</SPAN></SPAN></P>
<P class=MsoNormal><SPAN
style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">假设有以下三个一阶常微分方程式和其初始条件</SPAN><SPAN
lang=EN-US><SPAN style="mso-spacerun: yes">
</SPAN></SPAN></P>
<P class=MsoNormal><SPAN lang=EN-US>y'=3x2, y(2)=0.5<SPAN
style="mso-spacerun: yes"> </SPAN></SPAN></P>
<P class=MsoNormal><SPAN lang=EN-US>y'=2.x.cos(y)2, y(0)=0.25<SPAN
style="mso-spacerun: yes">
</SPAN></SPAN></P>
<P class=MsoNormal><SPAN lang=EN-US>y'=3y+exp(2x), y(0)=3<SPAN
style="mso-spacerun: yes"> </SPAN></SPAN></P>
<P class=MsoNormal><SPAN
style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">对应上述常微分方程式的符号运算式为：</SPAN><SPAN
lang=EN-US><SPAN style="mso-spacerun: yes">
</SPAN></SPAN></P>
<P class=MsoNormal><SPAN lang=EN-US>>>soln_1 = dsolve('Dy =
3*x^2','y(2)=0.5')<SPAN
style="mso-spacerun: yes">
</SPAN></SPAN></P>
<P class=MsoNormal><SPAN lang=EN-US>ans= x^3-7.500000000000000<SPAN
style="mso-spacerun: yes"> </SPAN></SPAN></P>
<P class=MsoNormal><SPAN lang=EN-US>>>ezplot(soln_1,[2,4]) % </SPAN><SPAN
style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">看看</SPAN><SPAN
style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">这个函数的</SPAN><SPAN
lang=EN-US></SPAN><SPAN
style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">长相</SPAN><SPAN
lang=EN-US><SPAN style="mso-spacerun: yes">
</SPAN></SPAN></P>
<P class=MsoNormal> </P>
<P class=MsoNormal><SPAN lang=EN-US>>>soln_2 = dsolve('Dy =
2*x*cos(y)^2','y(0) = pi/4')<SPAN
style="mso-spacerun: yes">
</SPAN></SPAN></P>
<P class=MsoNormal><SPAN lang=EN-US>ans= atan(x^2+1)<SPAN
style="mso-spacerun: yes"> </SPAN></SPAN></P>
<P class=MsoNormal><SPAN lang=EN-US>>>soln_3 = dsolve('Dy = 3*y +
exp(2*x)',' y(0) = 3')<SPAN
style="mso-spacerun: yes">
</SPAN></SPAN></P>
<P class=MsoNormal><SPAN lang=EN-US>ans= -exp(2*x)+4*exp(3*x)<SPAN
style="mso-spacerun: yes"> </SPAN></SPAN></P>
<P class=MsoNormal> </P>
<P class=MsoNormal><B style="mso-bidi-font-weight: normal"><SPAN lang=EN-US
style="FONT-SIZE: 14pt; mso-bidi-font-size: 10.0pt">2</SPAN></B><B
style="mso-bidi-font-weight: normal"><SPAN
style="FONT-SIZE: 14pt; FONT-FAMILY: 宋体; mso-bidi-font-size: 10.0pt; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">．</SPAN></B><B
style="mso-bidi-font-weight: normal"><SPAN lang=EN-US
style="FONT-SIZE: 14pt; mso-bidi-font-size: 10.0pt">4</SPAN></B><B
style="mso-bidi-font-weight: normal"><SPAN
style="FONT-SIZE: 14pt; FONT-FAMILY: 宋体; mso-bidi-font-size: 10.0pt; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">非线性方程式的实根</SPAN></B><B
style="mso-bidi-font-weight: normal"><SPAN lang=EN-US
style="FONT-SIZE: 14pt; mso-bidi-font-size: 10.0pt"><SPAN
style="mso-spacerun: yes"> </SPAN><O:P></O:P></SPAN></B></P>
<P class=MsoNormal><SPAN lang=EN-US><SPAN
style="mso-spacerun: yes"> </SPAN><SPAN
style="mso-spacerun: yes"> </SPAN></SPAN><SPAN
style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">要求任一方程式的根有三步骤：</SPAN><SPAN
lang=EN-US><SPAN style="mso-spacerun: yes"> </SPAN></SPAN></P>
<P class=MsoNormal><SPAN lang=EN-US><SPAN
style="mso-spacerun: yes"> </SPAN></SPAN><SPAN
style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">先定义方程式。要注意必须将方程式安排成</SPAN><SPAN
lang=EN-US> f(x)=0 </SPAN><SPAN
style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">的形态，例如一方程式为</SPAN><SPAN
lang=EN-US>sin(x)=3</SPAN><SPAN
style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">，</SPAN><SPAN
lang=EN-US> </SPAN></P>
<P class=MsoNormal><SPAN
style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">则该方程式应表示为</SPAN><SPAN
lang=EN-US> f(x)=sin(x)-3</SPAN><SPAN
style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">。可以</SPAN><SPAN
lang=EN-US> m-file </SPAN><SPAN
style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">定义方程式。</SPAN><SPAN
lang=EN-US><SPAN style="mso-spacerun: yes"> </SPAN></SPAN></P>
<P class=MsoNormal><SPAN lang=EN-US><SPAN
style="mso-spacerun: yes"> </SPAN></SPAN><SPAN
style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">代入适当范围的</SPAN><SPAN
lang=EN-US> x, y(x) </SPAN><SPAN
style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">值，将该函数的分布图画出，藉以了解该方程式的「长相」。</SPAN><SPAN
lang=EN-US><SPAN style="mso-spacerun: yes"> </SPAN></SPAN></P>
<P class=MsoNormal><SPAN lang=EN-US><SPAN
style="mso-spacerun: yes"> </SPAN></SPAN><SPAN
style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">由图中决定</SPAN><SPAN
lang=EN-US>y(x)</SPAN><SPAN
style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">在何处附近</SPAN><SPAN
lang=EN-US>(x0)</SPAN><SPAN
style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">与</SPAN><SPAN
lang=EN-US> x </SPAN><SPAN
style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">轴相交，以</SPAN><SPAN
lang=EN-US>fzero</SPAN><SPAN
style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">的语法</SPAN><SPAN
lang=EN-US>fzero('function',x0) </SPAN><SPAN
style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">即可求出在</SPAN><SPAN
lang=EN-US> x0</SPAN><SPAN
style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">附近的根，其中</SPAN><SPAN
lang=EN-US> function </SPAN><SPAN
style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">是先前已定义的函数名称。如果从函数分布图看出根不只一个，则须再代入另一个在根附近的</SPAN><SPAN
lang=EN-US> x0</SPAN><SPAN
style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">，再求出下一个根。</SPAN><SPAN
lang=EN-US><SPAN style="mso-spacerun: yes"> </SPAN></SPAN></P>
<P class=MsoNormal><SPAN lang=EN-US><SPAN
style="mso-spacerun: yes"> </SPAN></SPAN><SPAN
style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">以下分别介绍几数个方程式，来说明如何求解它们的根。</SPAN><SPAN
lang=EN-US><SPAN style="mso-spacerun: yes"> </SPAN></SPAN></P>
<P class=MsoNormal><SPAN lang=EN-US><SPAN
style="mso-spacerun: yes"> </SPAN></SPAN><SPAN
style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">例一、方程式为</SPAN><SPAN
lang=EN-US><SPAN style="mso-spacerun: yes"> </SPAN></SPAN></P>
<P class=MsoNormal><SPAN lang=EN-US><SPAN
style="mso-spacerun: yes"> </SPAN>sin(x)=0<SPAN
style="mso-spacerun: yes"> </SPAN></SPAN></P>
<P class=MsoNormal><SPAN lang=EN-US><SPAN
style="mso-spacerun: yes"> </SPAN></SPAN><SPAN
style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">我们知道上式的根有</SPAN>
<SPAN
style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">，求根方式如下：</SPAN><SPAN
lang=EN-US><SPAN style="mso-spacerun: yes"> </SPAN></SPAN></P>
<P class=MsoNormal><SPAN lang=EN-US>>> r=fzero('sin',3) % </SPAN><SPAN
style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">因为</SPAN><SPAN
lang=EN-US>sin(x)</SPAN><SPAN
style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">是内建函数，其名称为</SPAN><SPAN
lang=EN-US>sin</SPAN><SPAN
style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">，因此无须定义它</SPAN><SPAN
lang=EN-US>,</SPAN><SPAN
style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">选择</SPAN><SPAN
lang=EN-US> x=3 </SPAN><SPAN
style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">附近求根</SPAN><SPAN
lang=EN-US><SPAN style="mso-spacerun: yes"> </SPAN></SPAN></P>
<P class=MsoNormal><SPAN lang=EN-US><SPAN style="mso-spacerun: yes">
</SPAN>r=3.1416<SPAN style="mso-spacerun: yes"> </SPAN></SPAN></P>
<P class=MsoNormal><SPAN lang=EN-US>>> r=fzero('sin',6) % </SPAN><SPAN
style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">选择</SPAN><SPAN
lang=EN-US> x=6 </SPAN><SPAN
style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">附近求根</SPAN><SPAN
lang=EN-US><SPAN style="mso-spacerun: yes"> </SPAN></SPAN></P>
<P class=MsoNormal><SPAN lang=EN-US>r = 6.2832<SPAN
style="mso-spacerun: yes"> </SPAN></SPAN></P>
<P class=MsoNormal><SPAN lang=EN-US><SPAN
style="mso-spacerun: yes"> </SPAN></SPAN><SPAN
style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">例二、方程式为</SPAN><SPAN
lang=EN-US>MATLAB </SPAN><SPAN
⌨️ 快捷键说明
复制代码
Ctrl + C
搜索代码
Ctrl + F
全屏模式
F11
切换主题
Ctrl + Shift + D
显示快捷键
?
增大字号
Ctrl + =
减小字号
Ctrl + -