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{\cs79\f0\fs24\cf2\i \b0 \ul0 Примечание
}
{. Здесь везде *) - означает задание исходных данных пользователем или 
}
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{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{передачу в последующий алгоритм предыдущего выражения.
}
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{\cs16\f2\fs24\cf2\i0 \b \ul0 F:=-4*y/(1+y^2)*diff(u(x,y), x)+2*y/(1+y^2)*diff(u
(x,y), y);
}
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{\pard\plain
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{Приведенные выше данные позволяют записать само линейное дифференциальное урав-
}
\par}
{\pard\plain
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{нение с частными производными  
}
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{\cs16\f2\fs24\cf2\i0 \b \ul0 de:=a[1,1]*diff(u(x,y), x$2)+2*a[1,2]*diff(diff(u(
x,y),x),y)+a[2,2]*diff(u(x,y), y$2)+F=0;
}
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{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{*) Введем обозначения, указывая на то, относительно какой переменной заданы нач
аль-
}
\par}
{\pard\plain
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{ные условия (
}
{\cs80\f0\fs24\cf2\i0 \b \ul0 выберите 
}
{обозначение
}
{\cs84\f0\fs24\cf2\i0 \b \ul0   
}
{\cs81\f0\fs24\cf2\i \b \ul0 х0
}
{\cs82\f0\fs24\cf2\i0 \b \ul0  
}
{или
}
{\cs101\f0\fs24\cf2\i0 \b \ul0  
}
{\cs83\f0\fs24\cf2\i \b \ul0 у0
}
{ - это важно для дальнейшего построения 
}
\par}
{\pard\plain
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{алгоритма!) 
}
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{\pard\plain
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{\cs16\f2\fs24\cf2\i0 \b \ul0 y0:=1:p:=y0: u0:=x: du0:=0: 
}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{и зададим начальные условия:
}
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{\cs16\f2\fs24\cf2\i0 \b \ul0 if p=x0 then yx0:=u(x0,y)=u0,D[1](u)(x0,y)=du0:
}
\par}
{\pard\plain
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{\cs16\f2\fs24\cf2\i0 \b \ul0 else yx0:=u(x,y0)=u0,D[2](u)(x,y0)=du0 fi:
}
\par}
{\pard\plain
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{\cs16\f2\fs24\cf2\i0 \b \ul0 yx0;
}
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}
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{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{*) Здесь оператор 
}
{\cs18\f0\fs24\cf2\i \b0 \ul0 D
}
{ имеет следующую семантику
}
\par}
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{\cs16\f2\fs24\cf2\i0 \b \ul0 D[2](u)(x,y0)=du0: convert(%,diff);
}
\par}
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в
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}
{ проведем выбор типа уравнения с 
}
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{   
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}
{Ответ:
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}
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}
{из переменных 
}
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}
{ или 
}
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}
{ функцией 
}
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\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{*) Поэтому будем считать 
}
{\cs92\f0\fs24\cf2\i \b0 \ul0 x
}
{ функцией от 
}
{\cs95\f0\fs24\cf2\i \b0 \ul0 y
}
{, то есть 
}
{\cs93\f0\fs24\cf2\i \b0 \ul0 x
}
{(
}
{\cs96\f0\fs24\cf2\i \b0 \ul0 y
}
{). Для этого проведем замену 
}
{\cs97\f0\fs24\cf2\i \b0 \ul0 x 
}
{на
}
{\cs94\f0\fs24\cf2\i \b0 \ul0  
}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{\cs100\f0\fs24\cf2\i \b0 \ul0 x
}
{(
}
{\cs98\f0\fs24\cf2\i \b0 \ul0 y
}
{) в коэффициентах a[1,2], a[1,1]  и 
}
{\cs99\f0\fs24\cf2\i \b0 \ul0 qdii
}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{\cs16\f2\fs24\cf2\i0 \b \ul0 A[1,2]:=subs(x=x(y),a[1,2]);A[1,1]:=subs(x=x(y),a[
1,1]);Qdii:=subs(x=x(y),qdii);
}
\par}
{\pard\plain
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{\pict \dn13 \picw2711 \pich689\wmetafile8
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}
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{\pard\plain
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{\pict \dn6 \picw2799 \pich565\wmetafile8
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00000000
}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{Такая замена необходима для правильной работы Maple при решении характеристичес
-
}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{кого уравнения. С учетом проведенной замены запишем характеристическое уравнени
е,
}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{распадающееся на два,  
}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{\cs16\f2\fs24\cf2\i0 \b \ul0 yx1:=diff(x(y),y)=A[1,1]/(A[1,2]+Qdii);
}
\par}
{\pard\plain
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{\pict \dn23 \picw4747 \pich1036\wmetafile8
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0000
}
\par}
{\pard\plain
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{\cs16\f2\fs24\cf2\i0 \b \ul0 yx2:=diff(x(y),y)=A[1,1]/(A[1,2]-Qdii);
}
\par}
{\pard\plain
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{\pict \dn23 \picw4198 \pich1036\wmetafile8
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}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{для каждого из которых получим решение. Найдем  решение уравнения 
}
{\cs21\f0\fs24\cf2\i \b0 \ul0 ух
}
{1
}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{\cs16\f2\fs24\cf2\i0 \b \ul0 dsolve(yx1);
}
\par}
{\pard\plain
\s186 \f0 \fs24 \cf2 \i0 \b0 \ul0 \qc \sl360 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{\pict \dn21 \picw4092 \pich1106\wmetafile8
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}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{Найдем  решение уравнения 
}
{\cs46\f0\fs24\cf2\i \b0 \ul0 ух
}
{2
}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{\cs16\f2\fs24\cf2\i0 \b \ul0 dsolve(yx2);
}
\par}
{\pard\plain
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}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{Полученные решения надо записать в виде общего интеграла, оставляя системную ко
н-
}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{станту _
}
{\cs102\f0\fs24\cf2\i \b0 \ul0 C1
}
{ в одной части уравнения, а всё остальноё перенося в другую. 
}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{*) Итак, записываем новые переменные, снова возвращаясь к написанию 
}
{\cs19\f0\fs24\cf2\i \b0 \ul0 х 
}
{и 
}
{\cs20\f0\fs24\cf2\i \b0 \ul0 у
}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{\cs16\f2\fs24\cf2\i0 \b \ul0 xi:=(x,y)->x-2*y^3/3; eta:=(x,y)->x+2*y;
}
\par}
{\pard\plain
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\par}
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\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{*) Зависимости 
}
{\cs26\f0\fs24\cf2\i \b0 \ul0 xi
}
{(x,y) и 
}
{\cs27\f0\fs24\cf2\i \b0 \ul0 eta
}
{(x,y) должны допускать взаимно однозначное преобразование
}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{одних переменных в другие, то есть  
}
{\cs28\f0\fs24\cf2\i \b0 \ul0 x
}
{(
}
{\cs103\f0\fs24\cf2\i \b0 \ul0 xi
}
{,
}
{\cs104\f0\fs24\cf2\i \b0 \ul0 eta
}
{)   
}
{\cs29\f0\fs24\cf2\i \b0 \ul0 y
}
{(
}
{\cs105\f0\fs24\cf2\i \b0 \ul0 xi
}
{,
}
{\cs106\f0\fs24\cf2\i \b0 \ul0 eta
}
{). Поэтому в общем случае последние
}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{зависимости следует находить из системы двух уравнений:
}
\par}
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070000002105010078009902C6180700000021050100790099025D1A030000000000
}
\par}
{\pard\plain
\s187 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl360 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{\pict \dn31 \picw15274 \pich565\wmetafile8
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040000002D0100000700000021050100360085011331040000002D01020007000000210502005F5A
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}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{Запишем решение в явном виде (правда, очень длинное)
}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{\cs16\f2\fs24\cf2\i0 \b \ul0 JJ:=allvalues(%): 
}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{И в случае необходимости произведем подстановку 
}
{\cs107\f0\fs24\cf2\i \b0 \ul0 x
}
{(
}
{\cs109\f0\fs24\cf2\i \b0 \ul0 xi
}
{,
}
{\cs110\f0\fs24\cf2\i \b0 \ul0 eta
}
{)   
}
{\cs108\f0\fs24\cf2\i \b0 \ul0 y
}
{(
}
{\cs111\f0\fs24\cf2\i \b0 \ul0 xi
}
{,
}
{\cs112\f0\fs24\cf2\i \b0 \ul0 eta
}
{) при переходе к 
}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{новым переменным.
}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{*) Чтобы это сделать в алгоритме, введем заранее подстановочные значения этих п
ере-
}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{менных, записывая их как 
}
{\cs57\f0\fs24\cf2\i \b0 \ul0 ур
}
{ и 
}
{\cs58\f0\fs24\cf2\i \b0 \ul0 хр
}
{, значения которых возьмём из 
}
{\cs113\f0\fs24\cf2\i \b0 \ul0 JJ
}
{.
}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{\cs16\f2\fs24\cf2\i0 \b \ul0 yp:= 1/2*(-6*xi+6*eta+2*(16+9*xi^2-18*xi*eta+9*eta
^2)^(1/2))^(1/3)-2/(-6*xi+6*eta+2*(16+9*xi^2-18*xi*eta+9*eta^2)^(1/2))^(1/3);   
                                                                  xp:= eta-(-6*x
i+6*eta+2*(16+9*xi^2-18*xi*eta+9*eta^2)^(1/2))^(1/3)+4/(-6*xi+6*eta+2*(16+9*xi^2
-18*xi*eta+9*eta^2)^(1/2))^(1/3);
}
\par}
{\pard\plain
\s187 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl360 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{\pict \dn152 \picw9665 \pich2780\wmetafile8
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3A03030000000000
}
\par}
{\pard\plain
\s187 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl360 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{\pict \dn123 \picw10061 \pich2239\wmetafile8
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0000000000009001000000000400001254696D6573204E657720526F6D616E00E8B1F377F1B1F377
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BF23070000002105010031006F063123070000002105010033006F060624040000002D0101000800
000021050300202B2000D1053A03030000000000
}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{*) и аналогично - новых переменных 
}
{\cs63\f0\fs24\cf2\i \b0 \ul0 xip
}
{  
}
{\cs64\f0\fs24\cf2\i \b0 \ul0 etap, 
}
{когда это потребуется
}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{\cs16\f2\fs24\cf2\i0 \b \ul0 xip:=xi(x,y); etap:=eta(x,y);
}
\par}
{\pard\plain
\s186 \f0 \fs24 \cf2 \i0 \b0 \ul0 \qc \sl360 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{\pict \dn21 \picw3206 \pich1106\wmetafile8
0100090000031A01000005001C00000000000400000003010800050000000B020000000005000000
0C025204860C050000002E0118000000050000000201010000000500000009020000FF001C000000
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030007000000210501003300D400470B040000002D01000007000000210501003300A203A7090300
00000000
}
\par}
{\pard\plain
\s186 \f0 \fs24 \cf2 \i0 \b0 \ul0 \qc \sl360 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{\pict \dn7 \picw3153 \pich495\wmetafile8
010009000003C200000003001C00000000000400000003010800050000000B020000000005000000
0C02EF01510C050000002E0118000000050000000201010000000500000009020000FF001C000000
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040000002D0101000700000021050200797E3F012C0A030000000000
}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{Напомним, что переход к новым переменным должен привести к уравнению
}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{\pict \dn64 \picw14686 \pich1176\wmetafile8
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\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{Так как в уравнении гиперболического типа при переходе к новым переменным 
}
{\cs131\f0\fs24\cf2\i \b0 \ul0 xi,
}
{  
}
{\cs132\f0\fs24\cf2\i \b0 \ul0 eta
}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{коэффициенты a1[1,1]=a1[2,2]=0, 
}
\par}
{\pard\plain
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{\cs16\f2\fs24\cf2\i0 \b \ul0 a1[1,1]:=0: a1[2,2]:=0:
}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{то нам надо найти только коэффициент a1[1,2] и выражение F1 
}
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\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{Вычислим значение коэффициента 
}
{\cs71\f0\fs24\cf2\i \b0 \ul0 a
}
{1[1,2]:
}
\par}
{\pard\plain
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{\cs16\f2\fs24\cf2\i0 \b \ul0 a1[1,2]:=factor(simplify(a1[1,2])); 
}
\par}
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\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{При этом  уравнение  de1 упрощается до вида
}
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{\cs16\f2\fs24\cf2\i0 \b \ul0 de1;
}
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{Где для 
}
{\cs23\f0\fs24\cf2\i \b0 \ul0 F
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{1 надо взять формулу
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{\pard\plain
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{Проведем необходимые замены (подстановки) в выражениии для 
}
{\cs30\f0\fs24\cf2\i \b0 \ul0 F 
}
{в строгой последова-
}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{тельности как изложено ниже, чтобы избежать нежелательных подстановок в аргумен
ты 
}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{функции u(x,y):
}
\par}
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{\cs16\f2\fs24\cf2\i0 \b \ul0 F:=subs(diff(u(x,y),x)=dux,F); F:=subs(diff(u(x,y)
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}
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\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{Заменяем переменные 
}
{\cs59\f0\fs24\cf2\i \b0 \ul0 х
}
{ и 
}
{\cs60\f0\fs24\cf2\i \b0 \ul0 у
}
{ на новые и по возможности упрощаем выражение 
}
{\cs61\f0\fs24\cf2\i \b0 \ul0 F
}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{\cs16\f2\fs24\cf2\i0 \b \ul0 F:=subs(x=xp,F): F:=subs(y=yp,F): F:=simplify(F); 
}
\par}
{\pard\plain
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}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{Теперь можно составить и вычислить выражение 
}
{\cs25\f0\fs24\cf2\i \b0 \ul0 F
}
{1:
}
\par}
{\pard\plain
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}
\par}
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}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{Полученное выражение также попытаемся упростить: 
}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{\cs16\f2\fs24\cf2\i0 \b \ul0 F1:=simplify(F1);
}
\par}
{\pard\plain
\s186 \f0 \fs24 \cf2 \i0 \b0 \ul0 \qc \sl360 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{\pict \dn7 \picw1488 \pich495\wmetafile8
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}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{Окончательно запишем полученное таким образом уравнение и, чтобы получить устой
-
}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{чивое решение, отстроимся от старых обозначений.
}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{\cs16\f2\fs24\cf2\i0 \b \ul0 de1:=simplify(de1); de1:=subs(xi=v,eta=w,de1);
}
\par}
{\pard\plain
\s186 \f0 \fs24 \cf2 \i0 \b0 \ul0 \qc \sl360 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
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1316040000002D0100000700000021050100300099025D1A030000000000
}
\par}
{\pard\plain
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}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{Используя опцию 
}
{\cs72\f0\fs24\cf2\i \b0 \ul0 pdsolve
}
{, проведем решение, которое в данном случае не зависит от 
}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{начальных условий 
}
{\cs47\f0\fs24\cf2\i \b0 \ul0 ух0
}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{\cs16\f2\fs24\cf2\i0 \b \ul0 ans:=pdsolve(\{de1\},u(v,w));
}
\par}
{\pard\plain
\s186 \f0 \fs24 \cf2 \i0 \b0 \ul0 \qc \sl360 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{\pict \dn7 \picw6097 \pich495\wmetafile8
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}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{Здесь _
}
{\cs115\f0\fs24\cf2\i \b0 \ul0 F
}
{1(
}
{\cs114\f0\fs24\cf2\i \b0 \ul0 w
}
{)  и _
}
{\cs116\f0\fs24\cf2\i \b0 \ul0 F
}
{2(
}
{\cs117\f0\fs24\cf2\i \b0 \ul0 v
}
{)  любые функции, композиция которых 
}
{\cs122\f0\fs24\cf2\i \b0 \ul0 u
}
{(
}
{\cs123\f0\fs24\cf2\i \b0 \ul0 v
}
{,
}
{\cs124\f0\fs24\cf2\i \b0 \ul0 w
}
{)= _
}
{\cs119\f0\fs24\cf2\i \b0 \ul0 F
}
{1(
}
{\cs118\f0\fs24\cf2\i \b0 \ul0 w
}
{) + _
}
{\cs120\f0\fs24\cf2\i \b0 \ul0 F
}
{2(
}
{\cs121\f0\fs24\cf2\i \b0 \ul0 v
}
{) 
}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{удовлетворяет уравнению 
}
{\cs125\f0\fs24\cf2\i \b0 \ul0 de1
}
{. Подставим полученное решение в дифференциальное 
}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{уравнение 
}
{\cs48\f0\fs24\cf2\i \b0 \ul0 de
}
{1 и убедимся в том, что оно обращает его в тождество
}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{\cs16\f2\fs24\cf2\i0 \b \ul0 pdetest(ans,de1);
}
\par}
{\pard\plain
\s186 \f0 \fs24 \cf2 \i0 \b0 \ul0 \qc \sl360 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{\pict \dn7 \picw443 \pich495\wmetafile8
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}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{Как и следовало ожидать, получили ноль. Здесь по-прежнему следует отождествлять
 
}
{\cs31\f0\fs24\cf2\i \b0 \ul0 v
}
{ и 
}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{\cs32\f0\fs24\cf2\i \b0 \ul0 w
}
{ с новыми переменными 
}
{\cs33\f0\fs24\cf2\i \b0 \ul0 xi 
}
{и  
}
{\cs34\f0\fs24\cf2\i \b0 \ul0 eta
}
{: 
}
{\cs35\f0\fs24\cf2\i \b0 \ul0 v
}
{=
}
{\cs36\f0\fs24\cf2\i \b0 \ul0 xi
}
{,  
}
{\cs37\f0\fs24\cf2\i \b0 \ul0 w
}
{=
}
{\cs38\f0\fs24\cf2\i \b0 \ul0 eta
}
{. В полученном решении 
}
{\cs39\f0\fs24\cf2\i \b0 \ul0 ans 
}
{неизвестны-
}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{ми являются произвольные функции _
}
{\cs49\f0\fs24\cf2\i \b0 \ul0 F
}
{1(
}
{\cs51\f0\fs24\cf2\i \b0 \ul0 v
}
{) и _
}
{\cs50\f0\fs24\cf2\i \b0 \ul0 F
}
{2(
}
{\cs52\f0\fs24\cf2\i \b0 \ul0 w
}
{), которые ищутся из начальных усло-
}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{вий 
}
{\cs40\f0\fs24\cf2\i \b0 \ul0 ух0
}
{. Для их нахождения, применяя начальные условия 
}
{\cs62\f0\fs24\cf2\i \b0 \ul0 ух0
}
{, в 
}
{\cs41\f0\fs24\cf2\i \b0 \ul0 ans
}
{ перейдем к прежним 
}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{переменным 
}
{\cs42\f0\fs24\cf2\i \b0 \ul0 х
}
{ и 
}
{\cs43\f0\fs24\cf2\i \b0 \ul0 у
}
{, используя ранее найденные зависимости 
}
{\cs44\f0\fs24\cf2\i \b0 \ul0 xi
}
{(x,y) и 
}
{\cs45\f0\fs24\cf2\i \b0 \ul0 eta
}
{(x,y).
}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{\cs16\f2\fs24\cf2\i0 \b \ul0 ans:=subs(v=xi(x,y),w=eta(x,y),ans);
}
\par}
{\pard\plain
\s187 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl360 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{\pict \dn60 \picw12221 \pich1106\wmetafile8
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\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{*) В последнем выражении правую часть равенства  запишем  в виде функции 
}
{\cs53\f0\fs24\cf2\i \b0 \ul0 U
}
{(
}
{\cs54\f0\fs24\cf2\i \b0 \ul0 x
}
{,
}
{\cs55\f0\fs24\cf2\i \b0 \ul0 y
}
{)
}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{\cs16\f2\fs24\cf2\i0 \b \ul0 U:=(x,y)->_F2(x-2/3*y^3)+_F1(x+2*y);
}
\par}
{\pard\plain
\s186 \f0 \fs24 \cf2 \i0 \b0 \ul0 \qc \sl360 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{\pict \dn23 \picw7760 \pich1036\wmetafile8
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030000000000
}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{*) Определим производную от этой функции с одновременным вычислением её значени
я в 
}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{заданной точке 
}
{\cs65\f0\fs24\cf2\i \b0 \ul0 у0 
}
{(взять из "
}
{\cs126\f0\fs24\cf2\i \b0 \ul0 ух0
}
{" - как там записано: 
}
{\cs127\f0\fs24\cf2\i \b0 \ul0 D
}
{[2](
}
{\cs128\f0\fs24\cf2\i \b0 \ul0 u
}
{)(
}
{\cs129\f0\fs24\cf2\i \b0 \ul0 x
}
{,
}
{\cs130\f0\fs24\cf2\i \b0 \ul0 y0
}
{))
}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{\cs16\f2\fs24\cf2\i0 \b \ul0 dU:=D[2](U)(x,y0); 
}
{семантика оператора 
}
{\cs73\f0\fs24\cf2\i \b0 \ul0 D
}
{ пояснялась выше.
}
\par}
{\pard\plain
\s186 \f0 \fs24 \cf2 \i0 \b0 \ul0 \qc \sl360 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{\pict \dn23 \picw8362 \pich1036\wmetafile8
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}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{\cs16\f2\fs24\cf2\i0 \b \ul0 convert(D[2](U)(x,y0),diff);
}
\par}
{\pard\plain
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\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{*) После чего запишем по числу начальных условий  
}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{\cs16\f2\fs24\cf2\i0 \b \ul0 U(x,y0): dU:
}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{*) два уравнения
}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{\cs16\f2\fs24\cf2\i0 \b \ul0 yr1:=U(x,y0)=u0; yr2:=dU=du0;
}
\par}
{\pard\plain
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}
\par}
{\pard\plain
\s185 \f0 \fs36 \cf2 \i0 \b \ul0 \ql \sl240 \slmult1\sb160\sa80\pagebb0 \li0\fi0\ri0 
{
}
{\cs133\f0\fs24\cf2\i0 \b0 \ul0 
}
{\cs134\f0\fs24\cf2\i0 \b0 \ul0 Найдём функции _
}
{\cs138\f0\fs24\cf2\i \b0 \ul0 F1
}
{\cs139\f0\fs24\cf2\i0 \b0 \ul0 (
}
{\cs140\f0\fs24\cf2\i \b0 \ul0 w
}
{\cs141\f0\fs24\cf2\i0 \b0 \ul0 )  и  _
}
{\cs142\f0\fs24\cf2\i \b0 \ul0 F2
}
{\cs143\f0\fs24\cf2\i0 \b0 \ul0 (
}
{\cs144\f0\fs24\cf2\i \b0 \ul0 v
}
{\cs145\f0\fs24\cf2\i0 \b0 \ul0 )
}
{\cs135\f0\fs24\cf2\i \b0 \ul0 
}
{\cs136\f0\fs24\cf2\i0 \b0 \ul0  и само решение.
}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{*) Введем в этих уравнениях их общий аргумент 
}
{\cs137\f0\fs24\cf2\i \b0 \ul0 z
}
{=
}
{\cs146\f0\fs24\cf2\i \b0 \ul0 x
}
{+2, проводя подстановку
}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{\cs16\f2\fs24\cf2\i0 \b \ul0 yr1:=subs(x+2=z,x=z-2,yr1); yr2:=subs(x+2=z,x-2/3=
z-8/3,yr2);
}
\par}
{\pard\plain
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}
\par}
{\pard\plain
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}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{Далее решаем эту систему двух уравнений программными средствами Maple:
}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{1) из 
}
{\cs147\f0\fs24\cf2\i \b0 \ul0 yr1
}
{ выразим _
}
{\cs148\f0\fs24\cf2\i \b0 \ul0 F1
}
{(
}
{\cs149\f0\fs24\cf2\i \b0 \ul0 z
}
{)=
}
{\cs151\f0\fs24\cf2\i \b0 \ul0 f1
}
{ и продифференцируем по 
}
{\cs150\f0\fs24\cf2\i \b0 \ul0 z
}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{\cs16\f2\fs24\cf2\i0 \b \ul0 f1:=solve(yr1,_F1(z)); df1:=diff(f1,z);
}
\par}
{\pard\plain
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}
\par}
{\pard\plain
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{\pict \dn23 \picw4871 \pich1036\wmetafile8
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}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{2) последнее 
}
{\cs157\f0\fs24\cf2\i \b0 \ul0 df1
}
{ подставим в 
}
{\cs152\f0\fs24\cf2\i \b0 \ul0 yr2
}
{ и разрешим его относительно 
}
{\cs153\f0\fs24\cf2\i \b0 \ul0 D
}
{(_
}
{\cs154\f0\fs24\cf2\i \b0 \ul0 F2
}
{)(
}
{\cs155\f0\fs24\cf2\i \b0 \ul0 z
}
{-8/3)=
}
{\cs156\f0\fs24\cf2\i \b0 \ul0 f2
}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{\cs16\f2\fs24\cf2\i0 \b \ul0 yr3:=subs(D(_F1)(z)=df1,yr2); f2:=solve(yr3,D(_F2)
(z-8/3));
}
\par}
{\pard\plain
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}
\par}
{\pard\plain
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{\pict \dn23 \picw1364 \pich1036\wmetafile8
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}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{3) проинтегрируем 
}
{\cs158\f0\fs24\cf2\i \b0 \ul0 f2
}
{ и найдём _
}
{\cs159\f0\fs24\cf2\i \b0 \ul0 F2
}
{(
}
{\cs160\f0\fs24\cf2\i \b0 \ul0 z
}
{z), где 
}
{\cs161\f0\fs24\cf2\i \b0 \ul0 zz
}
{=
}
{\cs162\f0\fs24\cf2\i \b0 \ul0 z
}
{-8/3
}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{\cs16\f2\fs24\cf2\i0 \b \ul0 _F2(zz):=int(f2, zz);
}
\par}
{\pard\plain
\s186 \f0 \fs24 \cf2 \i0 \b0 \ul0 \qc \sl360 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{\pict \dn23 \picw2587 \pich1036\wmetafile8
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}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{4) теперь можем найти _
}
{\cs163\f0\fs24\cf2\i \b0 \ul0 F1
}
{(
}
{\cs164\f0\fs24\cf2\i \b0 \ul0 z
}
{) из 
}
{\cs165\f0\fs24\cf2\i \b0 \ul0 yr1
}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{\cs16\f2\fs24\cf2\i0 \b \ul0 yr4:=subs(_F2(z-8/3)=(z-8/3)/2,yr1); _F1(z):=solve
(yr4,_F1(z));
}
\par}
{\pard\plain
\s186 \f0 \fs24 \cf2 \i0 \b0 \ul0 \qc \sl360 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{\pict \dn23 \picw5263 \pich1036\wmetafile8
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}
\par}
{\pard\plain
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{\pict \dn23 \picw2977 \pich1036\wmetafile8
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}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{Таким образом 
}
{\cs166\f0\fs24\cf2\i0 \b0 \ul0 функции _
}
{\cs169\f0\fs24\cf2\i \b0 \ul0 F1
}
{\cs170\f0\fs24\cf2\i0 \b0 \ul0 (
}
{\cs171\f0\fs24\cf2\i \b0 \ul0 w
}
{\cs172\f0\fs24\cf2\i0 \b0 \ul0 )  и  _
}
{\cs173\f0\fs24\cf2\i \b0 \ul0 F2
}
{\cs174\f0\fs24\cf2\i0 \b0 \ul0 (
}
{\cs175\f0\fs24\cf2\i \b0 \ul0 v
}
{\cs176\f0\fs24\cf2\i0 \b0 \ul0 )
}
{\cs167\f0\fs24\cf2\i \b0 \ul0 
}
{\cs168\f0\fs24\cf2\i0 \b0 \ul0  найдены и мы можем записать решение
}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{U(x,y)=
}
{\cs177\f0\fs24\cf2\i0 \b0 \ul0 _
}
{\cs178\f0\fs24\cf2\i \b0 \ul0 F1
}
{\cs179\f0\fs24\cf2\i0 \b0 \ul0 (z)+_
}
{\cs181\f0\fs24\cf2\i \b0 \ul0 F2
}
{\cs182\f0\fs24\cf2\i0 \b0 \ul0 (zz)
}
{\cs180\f0\fs24\cf2\i \b0 \ul0 , 
}
{\cs183\f0\fs24\cf2\i0 \b0 \ul0 подставляя вместо z=x+2*y, zz=x-2/3*y^3. В итоге
 получим
}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{\cs16\f2\fs24\cf2\i0 \b \ul0 U:=(x,y)->(x+2*y)/2-2/3+(x-2/3*y^3)/2;
}
\par}
{\pard\plain
\s186 \f0 \fs24 \cf2 \i0 \b0 \ul0 \qc \sl360 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{\pict \dn23 \picw5333 \pich1036\wmetafile8
0100090000038F01000005001C00000000000400000003010800050000000B020000000005000000
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}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{Итак, решение найдено, остается только его проверить.
}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{\cs69\f0\fs24\cf2\i \b0 \ul0 Проверка решения.
}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{Подставим решение 
}
{\cs66\f0\fs24\cf2\i \b0 \ul0 U
}
{(
}
{\cs67\f0\fs24\cf2\i \b0 \ul0 x
}
{,
}
{\cs68\f0\fs24\cf2\i \b0 \ul0 y
}
{)  в исходное дифференциальное уравнение 
}
{\cs74\f0\fs24\cf2\i \b0 \ul0 de
}
{:
}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{\cs16\f2\fs24\cf2\i0 \b \ul0 re:=subs(u(x,y)=U(x,y),de): re;
}
\par}
{\pard\plain
\s186 \f0 \fs24 \cf2 \i0 \b0 \ul0 \qc \sl360 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{\pict \dn29 \picw6448 \pich1176\wmetafile8
010009000003DF01000006001C00000000000400000003010800050000000B020000000005000000
0C0298043019050000002E0118000000050000000201010000000500000009020000FF001C000000
FB025CFE0000000000009001000000020700001253796D626F6C0000450B0AA6FC880A00E8B1F377
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21050300202D20ED990252030800000021050300202B20ED99024F0A1C000000FB025CFE00000000
00009001000000000400001254696D6573204E657720526F6D616E00E8B1F377F1B1F3772040F577
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9001010000000400001254696D6573204E657720526F6D616E00E8B1F377F1B1F3772040F577490B
662D040000002D0102000700000021050200797E9902A90108000000FA0200000A0000000000FF00
040000002D01030005000000140224020D0505000000130224024F0A08000000FA02000000000000
00000000040000002D01040004000000F0010300040000002D010100070000002105010034008501
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000021050200797EE803D2071C000000FB02E8FE0000000000009001000000000400001254696D65
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21050100320037038C0908000000FA0200000A0000000000FF00040000002D010500050000001402
24020A0C05000000130224022416040000002D01040004000000F0010500040000002D0101000700
000021050100320085013F0C040000002D0102000700000021050200797E85017E0D040000002D01
0100070000002105010028008501910F0700000021050100290085016115040000002D0100000800
000021050300202D20ED85012911040000002D010100070000002105010031008501541004000000
2D0102000700000021050200797E8501E412040000002D01030007000000210501003200D4009E14
040000002D0100000800000021050300202B20EDE803800F040000002D0101000700000021050100
3100E803AB0E040000002D0102000700000021050200797EE8033B11040000002D01030007000000
2105010032003703F512040000002D010100070000002105010030009902DF17030000000000
}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{В правильности полученного решения можно убедиться, запустив процедуру упрощени
я, 
}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{в результате которой должны получить тождество:
}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{\cs16\f2\fs24\cf2\i0 \b \ul0 simplify(re); 
}
\par}
{\pard\plain
\s186 \f0 \fs24 \cf2 \i0 \b0 \ul0 \qc \sl360 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{\pict \dn7 \picw1099 \pich495\wmetafile8
0100090000037F00000002001C00000000000400000003010800050000000B020000000005000000
0C02EF014B04050000002E0118000000050000000201010000000500000009020000FF001C000000
FB025CFE0000000000009001000000020700001253796D626F6C0000170B0ACAFC880A00E8B1F377
F1B1F3772040F5774A0B664C040000002D0100000800000021050300203D20ED3F013F011C000000
FB025CFE0000000000009001000000000400001254696D6573204E657720526F6D616E00E8B1F377
F1B1F3772040F5774A0B664C040000002D010100070000002105010030003F016A00070000002105
010030003F01FA02030000000000
}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{*) Проведем также проверку на удовлетворение решения начальным условиям.
}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{Первому из них:
}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{\cs16\f2\fs24\cf2\i0 \b \ul0 simplify(U(x,y0));
}
\par}
{\pard\plain
\s186 \f0 \fs24 \cf2 \i0 \b0 \ul0 \qc \sl360 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{\pict \dn7 \picw655 \pich495\wmetafile8
0100090000035000000001001C00000000000400000003010800050000000B020000000005000000
0C02EF018F02050000002E0118000000050000000201010000000500000009020000FF001C000000
FB025CFE0000000000009001010000000400001254696D6573204E657720526F6D616E00E8B1F377
F1B1F3772040F577470B66B3040000002D0100000700000021050200787E3F016A00030000000000
}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{Второму:
}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{\cs16\f2\fs24\cf2\i0 \b \ul0 D[2](U)(x,y0); 
}
\par}
{\pard\plain
\s186 \f0 \fs24 \cf2 \i0 \b0 \ul0 \qc \sl360 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{\pict \dn7 \picw443 \pich495\wmetafile8
0100090000035000000001001C00000000000400000003010800050000000B020000000005000000
0C02EF01BB01050000002E0118000000050000000201010000000500000009020000FF001C000000
FB025CFE0000000000009001000000000400001254696D6573204E657720526F6D616E00E8B1F377
F1B1F3772040F5774A0B664E040000002D010000070000002105010030003F016A00030000000000
}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{Итак, проверка подтверждает правильность полученного решения.
}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{Дадим графическую интерпретацию полученного решения.
}
\par}
{\pard\plain
\s184 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 
{\cs16\f2\fs24\cf2\i0 \b \ul0 plot3d(U(x,y), x=-2..4, y=-4..4,axes=boxed,orienta
tion=[39,75]);
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{\pard\plain
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{\cs16\f2\fs24\cf2\i0 \b \ul0 
}
\par}
{\pard\plain
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> 
{\cs12\f2\fs24\cf1\i0 \b \ul0 
}
\par}
}