UIC：目录名直接使用UIC这个缩写

UIC/常微分方程/Chapter_1.tm

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+<TeXmacs|2.1.1>
+
+<style|beamer>
+
+<\body>
+  <screens|<\shown>
+    <doc-data|<doc-title|Ordinary Differential
+    Equations>|<doc-author|<author-data|<author-name|Yuliang Wang>>>>
+
+    <subsection|Motivation>
+
+    Suppose we deposite <math|u<rsub|0>> in a bank account, and the annual
+    interest rate is <math|r>. After <math|t> years,
+
+    <\enumerate>
+      <item>If the interest is compounded annualy, then the balance is
+
+      <\equation*>
+        u<around*|(|t|)>=u<rsub|0>*<around*|(|1+r|)><rsup|t>.
+      </equation*>
+
+      <item>If the interest is compounded monthly, then the balance is
+
+      <\equation*>
+        u<around*|(|t|)>=u<rsub|0>*<around*|(|1+<frac|r|12>|)><rsup|12*t>
+      </equation*>
+
+      <item>In general, if the interest is compounded <math|m> times a year,
+      then the balance is
+
+      <\equation*>
+        u<around*|(|t|)>=u<rsub|0>*<around*|(|1+<frac|r|m>|)><rsup|m*t>.
+      </equation*>
+
+      <item>Taking the limit as <math|m\<rightarrow\>\<infty\>>, then for any
+      fixed <math|t>, we have
+
+      <\equation*>
+        u<around*|(|t|)>=lim<rsub|m\<rightarrow\>\<infty\>><around*|(|1+<frac|r|m>|)><rsup|m*t>=lim<rsub|m\<rightarrow\>\<infty\>><with|color|red|<around*|(|1+<frac|r|m>|)>><rsup|<with|color|red|<frac|m|r>>*r*t>=e<rsup|r*t>.
+      </equation*>
+    </enumerate>
+
+    But we can obtain the same result by using a differential equation:
+
+    <\equation>
+      <with|color|blue|u<rprime|'><around*|(|t|)>=u<around*|(|t|)>*r>.
+    </equation>
+
+    We can solve it later and obtain the same result.
+
+    <\definition>
+      A <strong|ordinary differential equation> (ODE) about a function
+      <math|u<around*|(|t|)>> is an equation involving
+      <math|u<around*|(|t|)>> and its derivatives.
+    </definition>
+
+    <subsection|Classification of ODE>
+
+    Any ODE can be written in the abstract form
+
+    <\equation>
+      F<around*|(|u,u<rprime|'>,u<rprime|''>,\<ldots\>,u<rsup|<around*|(|n|)>>|)>=0.
+    </equation>
+
+    For example, Eq. (1) can be written as
+
+    <\equation*>
+      F<around*|(|u,u<rprime|'>|)>=r*u-u<rprime|'>=0.
+    </equation*>
+
+    In Eq. (2), <math|n> is the <strong|order> of the equation, i.e. the
+    order is the highest order derivative of <math|u> in the equation.\ 
+
+    So Eq. (1) is an ODE of order 1.\ 
+
+    If <math|F> is linear in terms of <math|u,u<rprime|'>,\<ldots\>,u<rsup|<around*|(|n|)>>>,
+    then the equation is called <strong|linear>. Otherwise it's called
+    <strong|nonlinear>. The general form of a linear ODE is
+
+    <\equation*>
+      a<rsub|n><around*|(|t|)>*u<rsup|<around*|(|n|)>>+a<rsub|n-1><around*|(|t|)>*u<rsup|<around*|(|n-1|)>>+\<cdots\>+a<rsub|1><around*|(|t|)>*u+a<rsub|0><around*|(|t|)>=0.
+    </equation*>
+
+    So Eq. (1) is linear. Examples of nonlinear equations:
+
+    <\equation*>
+      u<rprime|'>-u<rsup|2>=5,<space|1em>u*u<rprime|'>+5x=e<rsup|x>,<space|1em>u<rprime|'>=sin
+      u
+    </equation*>
+
+    <subsection|Solutions of an ODE>
+
+    A solution of an ODE
+
+    <\equation*>
+      F<around*|(|u,u<rprime|'>,u<rprime|''>,\<ldots\>,u<rsup|<around*|(|n|)>>|)>=0
+    </equation*>
+
+    is a function <math|u=\<phi\><around*|(|t|)>> satisfying the equation,
+    i.e.
+
+    <\equation*>
+      F<around*|(|\<phi\>,\<phi\><rprime|'>,\<phi\><rprime|''>,\<ldots\>,\<phi\><rsup|<around*|(|n|)>>|)><around*|(|t|)>=0.
+    </equation*>
+
+    <\example>
+      Can you give solutions of
+
+      <\equation*>
+        u<rprime|'>=2*u
+      </equation*>
+
+      possible solution: <math|u=e<rsup|2*t>>, in fact,
+      <math|u=c*e<rsup|2*t>> is a solution for any constant <math|c>.
+    </example>
+
+    <\example>
+      Can you give solutions of
+
+      <\equation*>
+        u<rprime|''>+4*u=0
+      </equation*>
+
+      A possible solution is <math|u=sin 2*t>, another solution is
+      <math|u<around*|(|t|)>=cos*2t.> In fact, any function
+
+      <\equation*>
+        u=a*sin 2*t+b*cos 2*t
+      </equation*>
+    </example>
+
+    \;
+
+    \;
+
+    \;
+
+    \;
+
+    \;
+
+    \;
+
+    \;
+
+    \;
+  </shown>>
+</body>
+
+<\initial>
+  <\collection>
+    <associate|magnification|2>
+    <associate|page-medium|papyrus>
+  </collection>
+</initial>
+
+<\references>
+  <\collection>
+    <associate|auto-1|<tuple|1|?>>
+    <associate|auto-2|<tuple|2|?>>
+    <associate|auto-3|<tuple|3|?>>
+  </collection>
+</references>
+
+<\auxiliary>
+  <\collection>
+    <\associate|toc>
+      <with|par-left|<quote|1tab>|1<space|2spc>Motivation
+      <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
+      <no-break><pageref|auto-1>>
+
+      <with|par-left|<quote|1tab>|2<space|2spc>Classification of ODE
+      <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
+      <no-break><pageref|auto-2>>
+
+      <with|par-left|<quote|1tab>|3<space|2spc>Solutions of an ODE
+      <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
+      <no-break><pageref|auto-3>>
+    </associate>
+  </collection>
+</auxiliary>