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