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北京师范大学：微积分和偏微分方程 第一次课件上传
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<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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\;
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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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|
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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>
|
||||
</collection>
|
||||
</auxiliary>
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|
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@ -0,0 +1,392 @@
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<TeXmacs|2.1.1>
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|
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<style|<tuple|projector|reddish|framed-theorems>>
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<\body>
|
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<screens|<\shown>
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<doc-data|<doc-title|Calculus (1003)>|<doc-author|<author-data|<author-name|Yuliang
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Wang>>>>
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<section|Functions and Models>
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<\definition>
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A function is a <strong|rule> to assign to a variable, say <math|x>, an
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<strong|unique> value, say <math|y>. In general, we can write
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|
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<\equation*>
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y=f<around*|(|x|)>.
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</equation*>
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|
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Here
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<\itemize>
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<item><math|f> is the name of the function
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<item><math|x> is called the <strong|independent variable>
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<item><math|y> is called the <strong|dependent variable>
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</itemize>
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|
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The set of values that <math|x> can take is called the <strong|domain>
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of the function. The set of all possible values of <math|y> constitute
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the <strong|range>.
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<math|f<around*|(|x|)>> is read <strong|f of x.>
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</definition>
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We can view a function as a machine, that takes the input <math|x>, and
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gives the output <math|y>.
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The <strong|graph> of a <with|ornament-border|0ln|function>
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<math|y=f<around*|(|x|)>> is the curve
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<\equation*>
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<around*|{|<around*|(|x,y|)>\<in\>\<bbb-R\><rsup|2>:y=f<around*|(|x|)>|}>.
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</equation*>
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<math|\<bbb-R\><rsup|2>>: the <math|x*y> plane
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<strong|Vertical Line Test>
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<with|gr-mode|<tuple|edit|document-at>|gr-frame|<tuple|scale|1cm|<tuple|0.490001gw|0.499999gh>>|gr-geometry|<tuple|geometry|1par|0.6par>|gr-color|red|<graphics||<with|arrow-end|\<gtr\>|<line|<point|-5.38727|-0.729059>|<point|-1.1991483732275|-0.672206374798462>>>|<with|arrow-end|\<gtr\>|<line|<point|-4.59765|-1.11439>|<point|-4.64818719252553|2.50521311339865>>>|<with|color|blue|<spline|<point|-5.03352|0.799636>|<point|-3.78908594815825|-0.192120385299103>|<point|-2.3993633469759|0.951242300219108>|<point|-1.42655752614825|0.818586961015338>>>|<with|color|blue|<math-at|y=f<around*|(|x|)>|<point|-1.43200936235893|0.422340771658782>>>|<with|color|blue|<math-at|x|<point|-1.08544|-0.842763>>>|<with|color|blue|<math-at|y|<point|-4.96404|2.45468>>>|<with|color|red|<line|<point|-2.85692609418133|2.10957038347715>|<point|-2.83166562165249|-1.14364229294602>>>|<with|color|red|<math-at|x=a|<point|-3.00579|-1.29517>>>|<with|color|red|<point|-2.84521|0.601097>>|<with|color|red|<math-at|<around*|(|a,b|)>|<point|-2.58887|0.486204>>>|<with|color|red|dash-style|11100|<line|<point|-2.84521|0.601097>|<point|-4.62122789064825|0.574318892063522>>>|<with|color|red|<math-at|b|<point|-4.47131|0.7831>>>|<with|color|red|<\document-at>
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This <with|color|blue|curve> represents the\
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graph of a function
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</document-at|<point|-4.93246|-1.80684>>>|<line|<point|0.481153|-0.783497>|<point|4.98511720203398|-0.78981355161437>>|<line|<point|1.10653|-1.40255>|<point|1.10021083963785|2.04648393898053>>|<with|color|blue|<spline|<point|0.658026|-0.00651536>|<point|2.88789945843152|-0.385530613088594>|<point|2.77419488197114|1.09262888089628>|<point|4.34079126875853|1.20633345735665>>>|<with|color|red|<line|<point|2.75524|1.89488>|<point|2.80577948654347|-1.50994253586341>>>|<with|color|red|<point|2.76722|1.08772>>|<with|color|red|<point|2.77621|0.482115>>|<with|color|red|<point|2.78921|-0.39335>>|<with|color|red|<math-at|x=a|<point|2.6668|-1.73103>>>|<with|color|red|<math-at|<around*|(|a,b|)>|<point|2.93843|-0.543454>>>|<with|color|red|<math-at|<around*|(|a,c|)>|<point|2.93843|0.517789>>>|<with|color|red|<math-at|<around*|(|a,d|)>|<point|2.87527|1.37057>>>|<with|color|red|<\document-at>
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This curve doesn't represent the
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graph of a function
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</document-at|<point|0.24111|-1.97108>>>>>
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<\em>
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<strong|piecewise defined function>
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</em>
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<\equation*>
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y=<choice|<tformat|<table|<row|<cell|x,>|<cell|x\<geqslant\>0>>|<row|<cell|-x>|<cell|x\<leqslant\>0>>>>>
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</equation*>
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This function can also be written as
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<\equation*>
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y=<around*|\||x|\|>.
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</equation*>
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<strong|symmetry> (even/odd functions)
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<strong|increasing/decreasing> functions
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<section|A catalog of essential functions>
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<\itemize>
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<item>linear functions
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<\equation*>
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y=m*x+b
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</equation*>
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where <math|m> is the <strong|slope>, and <math|b> is the
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<math|y>-<strong|intercept>.
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For a linear function, if <math|y<rsub|1>=m*x<rsub|1>+b,y<rsub|2>=m*x<rsub|2>+b>,
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then
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<\equation*>
|
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y<rsub|2>-y<rsub|1>=m*<around*|(|x<rsub|2>-x<rsub|1>|)>.
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</equation*>
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Graphing calculator: www.desmos.com
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<item>power functions
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<\equation*>
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y=x<rsup|a>
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</equation*>
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The property depends on the value of <math|a>.
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<\itemize>
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<item>If <math|a=2,4,6,\<cdots\>>, then it's an even function, and
|
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it's increasing function for <math|x\<geqslant\>0> and decreasing for
|
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<math|x\<leqslant\>0>.
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<item>If <math|a=1,3,5,\<cdots\>>, then it's an odd function, and
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it's increasing function for all <math|x>.
|
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|
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<item>If <math|a=-2,-4,-6,\<cdots\>>, then it's an even function, and
|
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it's decreasing function for <math|x\<gtr\>0> and increasing for
|
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<math|x\<less\>0>.
|
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|
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<item>If <math|a=-1,-3,-5,\<cdots\>>, then it's an odd function, and
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it's decreasing function for <math|x\<gtr\>0> or <math|x\<less\>0>.
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<item>If <math|a=<frac|1|2>,<frac|1|4>,<frac|1|6>,\<cdots\>>, then
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the function is defined only for <math|x\<geqslant\>0>, and it's an
|
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increasing function.
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|
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<item> If <math|a=<frac|1|3>,<frac|1|5>,<frac|1|7>,\<cdots\>>, then
|
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the function is defined for all <math|x>, and it's an increasing
|
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function.
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<item>other cases: read the slides or book
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</itemize>
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<item>polynomials
|
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|
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<\equation*>
|
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y=a<rsub|n>*x<rsup|n>+a<rsub|n-1>*x<rsup|n-1>+\<cdots\>+a<rsub|1>*x+a<rsub|0>,
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</equation*>
|
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|
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where <math|n=0,1,2,<gap|>>is called the <strong|degree> of the
|
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polynomial, and <math|a<rsub|i>>'s are called the
|
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<strong|coefficients>. For example, <math|y=3*x<rsup|2>-x+5> is a
|
||||
polynomial of degree 2, or we call it a <strong|quadratic polynomial>,
|
||||
and <math|y=x<rsup|3>+2x<rsup|2>-10> is a polynomial of degree 3, also
|
||||
called a <strong|cubic polynomial>.
|
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|
||||
The domain of any polynomial is <math|\<bbb-R\>=<around*|(|-\<infty\>,\<infty\>|)>>.
|
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|
||||
<item>rational functions
|
||||
|
||||
<\equation*>
|
||||
y=<frac|P<around*|(|x|)>|Q<around*|(|x|)>>,
|
||||
</equation*>
|
||||
|
||||
where <math|P,Q> are both polynomials. For example
|
||||
|
||||
<\equation*>
|
||||
y=<frac|x<rsup|2>-2x|4x<rsup|5>-5>
|
||||
</equation*>
|
||||
|
||||
The domain of a rational function is all <math|x> except where
|
||||
<math|Q<around*|(|x|)>=0>.
|
||||
|
||||
<item>algebraic functions: any function the can be obtained by
|
||||
addition, subtraction, multiplication, and raising to rational powers
|
||||
of <math|x>. For example
|
||||
|
||||
<\equation*>
|
||||
y=<frac|x<rsup|2>+3*<sqrt|x>|x<rsup|3/2>-4>+10*x<rsup|9>.
|
||||
</equation*>
|
||||
|
||||
<item>exponential functions
|
||||
|
||||
<\equation*>
|
||||
y=b<rsup|x>,
|
||||
</equation*>
|
||||
|
||||
where <math|b\<gtr\>0> is called the <strong|base>, <math|x> is called
|
||||
the <strong|exponent>.
|
||||
|
||||
<item>logarithmic functions
|
||||
|
||||
<\equation*>
|
||||
y=log<rsub|b>*x
|
||||
</equation*>
|
||||
|
||||
they are actually the <strong|inverse functions> of exponential
|
||||
functions, i.e.
|
||||
|
||||
<\equation*>
|
||||
y=log<rsub|b> x\<Longleftrightarrow\>x=b<rsup|y>
|
||||
</equation*>
|
||||
|
||||
<item>trigonometric functions
|
||||
|
||||
<\equation*>
|
||||
y=sin x,cos x,tan x,cot x,sec x,csc x
|
||||
</equation*>
|
||||
</itemize>
|
||||
|
||||
Elementary building blocks:\
|
||||
|
||||
<\enumerate>
|
||||
<item>power functions
|
||||
|
||||
<item>exponential functions
|
||||
|
||||
<item>logarithmic functions
|
||||
|
||||
<item>trigonometric functions
|
||||
</enumerate>
|
||||
|
||||
<section|New functions from old>
|
||||
|
||||
<\itemize>
|
||||
<item>translations (move the graph up/down, left/right)
|
||||
|
||||
<\itemize>
|
||||
<item>vertical translation
|
||||
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||||
<\equation*>
|
||||
y=f<around*|(|x|)>+c
|
||||
</equation*>
|
||||
|
||||
<item>horizontal translation
|
||||
|
||||
<\equation*>
|
||||
y=f<around*|(|x-c|)>
|
||||
</equation*>
|
||||
</itemize>
|
||||
|
||||
<item>stretching and reflecting
|
||||
|
||||
<\itemize>
|
||||
<item>vertical stretching/reflecting
|
||||
|
||||
<\equation*>
|
||||
y=c*f<around*|(|x|)>
|
||||
</equation*>
|
||||
|
||||
<item>horizontal stretching/reflecting
|
||||
|
||||
<\equation*>
|
||||
y=f<around*|(|<frac|x|c>|)>
|
||||
</equation*>
|
||||
</itemize>
|
||||
|
||||
<item>algebraic combinations
|
||||
|
||||
<\equation*>
|
||||
f+g,f-g,f*g,<frac|f|g>
|
||||
</equation*>
|
||||
|
||||
<item>composition
|
||||
|
||||
<\equation*>
|
||||
f<around*|(|g<around*|(|x|)>|)>=<with|color|red|f\<circ\>g><around*|(|x|)>
|
||||
</equation*>
|
||||
|
||||
read as <math|f> of <math|g> of <math|x>, or <math|f> composed with
|
||||
<math|g>
|
||||
|
||||
<\example>
|
||||
If <math|f<around*|(|x|)>=x<rsup|2>+3*x> and
|
||||
<math|g<around*|(|x|)>=sin x>. Then
|
||||
|
||||
<\itemize>
|
||||
<item><math|f\<circ\>g<around*|(|x|)>=f<around*|(|<with|color|red|g<around*|(|x|)>>|)>=<around*|(|<with|color|red|g<around*|(|x|)>>|)><rsup|2>+3*<with|color|red|g<around*|(|x|)>>=<around*|(|sin
|
||||
x|)><rsup|2>+3*sin x>
|
||||
|
||||
<item><math|g\<circ\>f<around*|(|x|)>=g<around*|(|f<around*|(|x|)>|)>=sin
|
||||
f<around*|(|x|)>=sin <around*|(|x<rsup|2>+3*x|)>.>
|
||||
|
||||
<item><math|f\<circ\>f<around*|(|x|)>=<around*|(|f<around*|(|x|)>|)><rsup|2>+3*f<around*|(|x|)>=<around*|(|x<rsup|2>+3*x|)><rsup|2>+3*<around*|(|x<rsup|2>+3*x|)>>
|
||||
|
||||
<item><math|g\<circ\>g<around*|(|x|)>=sin g<around*|(|x|)>=sin
|
||||
<around*|(|sin x|)>.>
|
||||
</itemize>
|
||||
|
||||
<\equation*>
|
||||
x<long-arrow|\<rubber-rightarrow\>|g>g<around*|(|x|)><draw-over||<with|gr-mode|<tuple|group-edit|move>|gr-arrow-end|\<gtr\>|<graphics|<with|arrow-end|\<gtr\>|<spline|<point|-0.669548|-0.0948158>|<point|0.0712844681466782|-0.505448757699781>|<point|0.638550580842532|-0.12444912976973>>>|<math-at|f\<circ\>g|<point|-0.161548652131134|-0.649382353921204>>>>|2cm><long-arrow|\<rubber-rightarrow\>|f>f<around*|(|g<around*|(|x|)>|)>
|
||||
</equation*>
|
||||
</example>
|
||||
|
||||
<item>
|
||||
|
||||
<item>
|
||||
|
||||
<\example>
|
||||
Write the function <math|F<around*|(|x|)>=log<rsub|2>
|
||||
<sqrt|x<rsup|2>+1>> as a composition of 3 functions.
|
||||
|
||||
Answer: Let <math|f<around*|(|x|)>=log<rsub|2>
|
||||
x,g<around*|(|x|)>=<sqrt|x>,h<around*|(|x|)>=x<rsup|2>+1>. Then
|
||||
|
||||
<\equation*>
|
||||
F=f\<circ\>g\<circ\>h
|
||||
</equation*>
|
||||
</example>
|
||||
|
||||
<item>Application example (compound interest): Suppose you deposit an
|
||||
amount of <math|u<rsub|0>> in an account with annual interest rate of
|
||||
<math|r>. The balance after <math|t> years, if the interest is
|
||||
compounded
|
||||
|
||||
<\itemize>
|
||||
<item>annualy, then
|
||||
|
||||
<\equation*>
|
||||
u<around*|(|t|)>=u<rsub|0>*<around*|(|1+r|)><rsup|t>
|
||||
</equation*>
|
||||
|
||||
<item>monthly, then
|
||||
|
||||
<\equation*>
|
||||
u<around*|(|t|)>=u<rsub|0>*<around*|(|1+<frac|r|12>|)><rsup|12*t>=u<rsub|0>*<around*|[|<around*|(|1+<frac|r|12>|)><rsup|12>|]><rsup|t>
|
||||
</equation*>
|
||||
|
||||
<item><math|n> times per year, then
|
||||
|
||||
<\equation*>
|
||||
u<around*|(|t|)>=u<rsub|0><around*|(|1+<frac|r|n>|)><rsup|n*t>
|
||||
</equation*>
|
||||
|
||||
<item>continuously, then
|
||||
|
||||
<\equation*>
|
||||
u<around*|(|t|)>=lim<rsub|n\<rightarrow\>\<infty\>>u<rsub|0><around*|(|1+<frac|r|n>|)><rsup|n*t>=lim<rsub|n\<rightarrow\>\<infty\>>u<rsub|0><around*|[|<around*|(|1+<frac|r|n>|)><rsup|n>|]><rsup|*t>=u<rsub|0>*e<rsup|r*t>,
|
||||
</equation*>
|
||||
|
||||
where <math|e=2.71828\<cdots\>> is called the <strong|natural base>.
|
||||
</itemize>
|
||||
|
||||
Here we are using the formula
|
||||
|
||||
<\equation*>
|
||||
<with|color|blue|lim<rsub|n\<rightarrow\>\<infty\>><around*|(|1+<frac|1|n>|)><rsup|n>=e>
|
||||
</equation*>
|
||||
</itemize>
|
||||
|
||||
\;
|
||||
|
||||
\;
|
||||
|
||||
\;
|
||||
|
||||
\;
|
||||
|
||||
\;
|
||||
|
||||
\;
|
||||
|
||||
\;
|
||||
</shown>>
|
||||
</body>
|
||||
|
||||
<\initial>
|
||||
<\collection>
|
||||
<associate|font-base-size|10>
|
||||
<associate|info-flag|minimal>
|
||||
<associate|magnification|2>
|
||||
<associate|ornament-shape|classic>
|
||||
<associate|page-medium|papyrus>
|
||||
<associate|par-columns|1>
|
||||
</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>
|
||||
<vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|1<space|2spc>Functions
|
||||
and Models> <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><vspace|0.5fn>
|
||||
|
||||
<vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|2<space|2spc>A
|
||||
catalog of essential functions> <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><vspace|0.5fn>
|
||||
|
||||
<vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|3<space|2spc>New
|
||||
functions from old> <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><vspace|0.5fn>
|
||||
</associate>
|
||||
</collection>
|
||||
</auxiliary>
|
||||
Loading…
Reference in New Issue