F(t,y,y,y,...,y (m) ) = 0. y(t) z(t) = y (m 1) (t) G(t,z,z ) = 0

Transcript

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5 t s q c '&1-.(&$ )- I+&f1*G, ), F-%ie,'. I+M&(' II,13, 1J($.3b+1, )- I+&f1*G,,. 1 +3'&\ y 1-.(&$ ), %, 'e'.*g,,'. I+M&(' II,13, ($.3b+.(&$ ), 1J 4 9>A _4 ( f,'. -$, M&$%.(&$ -$( -,G,$. ), t2 1, I+&f1*G, ), F-%ie ', +G *$, K 1 +,%i,+%i, ) J-$, I+(G (.(/, 4 X $' 1, %' -$()(G,$'(&$$,1 2 y(t) IR2 % J,'. 3b1,G,$. 1, %' 1&+' -, 1J&$ I,-.,Y,%.-,+ -$, '3I+.(&$ ),' /+(f1,' 2 ( 4, 4 ^ f(t,y) = g(t)h(y) X $' %, %' 2 '( h(y) 2 &$ I,-. 3%+(+, 1, I+&f1*G, ), F-%ie '&-' 1 M&+G, '-(/$., ^ y t 4 dy y h(y) = Z g(t)dt 4 4 t (,'. -$, M&$%.(&$ 1($3(+, ), (1,:('., 1&+' ), $&Gf+,-',' G 3.i&),' $1e.( -,' I &-+ 4 f y2 +3'&-)+, 1, I+&f1*G, Z#+(.(&$ ), 1 %&$.$., 4 (,'. -$, M&$%.(&$ $&$ 1($3(+, ), %, I+&f1*G,,'. I1-' b3$3+1,g,$. f &+)3 $-G 3+(\ j 4 f y2 -,G,$. 4 * * * #!( & " &(& " & " /$. ), )&$$,+ ),'.i3&+*g,' ) J,:('.,$%, ), '&1-.(&$ 2 (1,'. f &$ ), I+3%(',+ -,11,.eI, ), '&1-.(&$ &$ +,%i,+%i, 4 h &-+ %,1 2 &$ ($.+&)-(. 1,' )3d$(.(&$' '-(/$.,' 4 N OP QCRCSQ TETET #$ II,11, '&1-.(&$ 1&%1, )- I+&f1*G, ), F-%ie 4j 1 )&$$3, ) J-$ %&-I1, (I,y) &` I,'. -$ ($.,+/11, ), IR -(,'. /&('($b, ), t )$' I,. &` y,'. -$, M&$%.(&$ ), %1'', C 1 '-+ I.,11, -, ^ y(t ) = y, t I,y (t) = f(t,y(t)) 4 N OP QCRCSQ TETE #$ )(. -, 1 '&1-.(&$ 1&%1, (J,z) I+&1&$b, 1 '&1-.(&$ 1&%1, (I,y) '( &$ I J, t I,y(t) = z(t) ( ), I1-' 2 I J &$ )(. -, 1 '&1-.(&$ (J,z) I+&1&$b, '.+(%.,G,$. 1 '&1-.(&$ (I,y)4 4 N OP QCRCSQ TETE #$ )(. -, 1 '&1-.(&$ 1&%1, (I,y) )- I+&f1*G, ), F-%ie 4j,'. G :(G 1, '( (1 $ J,:('., I' ), '&1-.(&$ 1&%1, -( 1 I+&1&$b, '.+(%.,G,$. 4 N OP QCRCSQ TETE #$ )(. -, 1 '&1-.(&$ (I,y),'. -$, '&1-.(&$ b1&f1, )- I+&f1*G, ), F-%ie 4j )$' I '( (I,y),'. -$, '&1-.(&$ 1&%1,,. I = I 4

6 j > B>A _4 F&$'()3+&$' 1, I+&f1*G, ), F-%ie '-(/$. ^ y (t) = 2ty 2 (t),t IR y() = 1 F, I+&f1*G, )G,. -$, '&1-.(&$ b1&f1,,. -$, ',-1, )$' IR ^ 4 (IR,y(t) = 1 Z 1 + t 2) 4 4_,. (1 $ Je I' ) J-.+, '&1-.(&$ G :(G 1, 4 4 c, I+&f1*G, ), F-%ie '-(/$. ^ y (t) = 2ty 2 (t),t IR y() = 1 4 )G,. -$, '&1-.(&$ G :(G 1,,. -$, ',-1, )$' % J,'. 1 $ Je IR2 (] 1,1,y(t) = 1 t 2)4 "1 I' ), '&1-.(&$ b1&f1, ( 1J&$ %i&('(. %&G G, %&$)(.(&$ ($(.(1, K %, I+&f1*G, 1 )&$$3, 4 y( 2) = 12 (1,:('., -$, '&1-.(&$ G :(G 1,,. -$, ',-1, -(,'. (] 5, 5,y(t) = 1 5 t 2)4 " 1,'. f &$ ), +,G + -,+ -, 1J,$',Gf1, '-+ 1, -,1 ', )3d$(. 1,' '&1-.(&$' ) J-$ I+&f1*G,,'. )3I,$)$. -''( ), 1 %&$)(.(&$ ($(.(1, 4 j 4 c, I+&f1*G, ), F-%ie '-(/$. ^ y (t) = y 2 (t),t IR + y() = 1 4_ )G,. -$, '&1-.(&$ b1&f1,,. -$, ',-1, )$' -(,'. 1 -+,+ 2 (],+,y(t) = )4 1 + t IR2 (1 $ Je I' ), '&1-.(&$ b1&f1, G (' ',-1,G,$. -$, '&1-.(&$ G :(G 1, '-+ ] 1,+ 4 m, I+&f1*G, ), F-%ie '-(/$. 4 c ^ y (t) = 3 y(t),t,+ Z 4 4_j y() = )G,. %&G G, '&1-.(&$' b1&f1,' ^ Z m (,+,y(t) = ) 4 4_ 8t 3 (,+,y(t) = 27 ) Z 4 4_ 8t 3 (,+,y(t) = 27 ) Z 4 4_ g &-' I &-/&$' %&$'..,+ -, 1,' I+&f1*G,' ) J,:('.,$%,,. ) J-$(%(.3 ), '&1-.(&$' $, '&$. I' ),' -,'.(&$'.+(/(1,' 4 g &-' 11&$' I+3',$.,+ )$' 1 '-(., -,1 -,' -$' ),'.i3&+*g,' M&$)G,$.-: ) J,:('.,$%,,. ) J-$(%(.3 ),' '&1-.(&$' G &e,$$$. ),' iei &.i*',' '-+ f 4 (1/y) = y /y 2

7 t s * * %) " $ "(!" " $" &# % "& ' (OSD) > T ETET 7@9?( >@QS #$ '-II &', -, f,'. %&$.($-, )$' -$ /&('($b, )- I &($. (t,y ) I IR n 2 1&+' (1,:('., -$, ($.,+/11, /&('($b, ), )$',. -$, M&$%.(&$ J t I y C 1 (J ).,1' -, ^ y (t) = f(t,y(t)), t J, Z 4 4_ y(t ) = y c,.i3&+*g, ), F-%ie\ h,$& )&$$, )&$% -$ +3'-1.. ) J,:('.,$%, ) J-$, '&1-.(&$ 1&%1, (J,y)4 * +, #&(+ Z 2 _ 4 c %&$)(.(&$ ), %&$.($-(.3 ), f $, '-a. I' I &-+ I'',+ K -$, '&1-.(&$ b1&f1, &-' /,++&$' 4 g )$' 1 '-(., - J(1 M-. (G I &',+ ),' %&$)(.(&$' '-II13G,$.(+,' + %&$.+, 1, +3'-1.. -( '-(. 4 h 2 I,+G,. ) J''-+,+ 1J,:('.,$%, ) J-$, '&1-.(&$ G :(G 1, 4 ' (OSD) > T ETE.$. )&$$3, -$, '&1-.(&$ 1&%1, 2 (1,:('., - G &($' ZI' M&+%3G,$. -$( -, -$, '&1-.(&$ G :(G 1, -( 1 I+&1&$b, 4 * +, #&(+ Z 2 _ 4 #$ I- +,G + -,+ '-+ 1,',:,G I1,' )- I+3%3),$. -, 1 $&.(&$ ), '&1-.(&$,'..+*' )3\ I,$)$., ), 1J,$',Gf1, '-+ 1, -,1 &$ 1 %i,+%i, #$ %&$'.., -''( 4 -, 1&+' -, ),' '&1-.(&$' G :(G 1,' $&$ b1&f1,',:('.,$. %,1,'. '&-/,$. )- K -$, '($b-1+(.3 ),,.i3&+*g, '-(/$. 2 y4 c I+3%(', %,.., ($.-(.(&$ 4 ' (OSD) > T ETE #$ '-II &', -,,'. %&$.($-, '-+ f I IR n,. -,,'. ), 1 M&+G, I t,t +T]2 t,t +T t,+ 1&+' '(,'. -$, '&1-.(&$ G :(G 1, $&$ b1&f1, )- I+&f1*G, ), F-%ie (I,y) Z,'. ), 1 M&+G, 4 4j 2 " t,t 1,. y $ J,'. I' f &+$3 '-+ I 4 * +, #&(+ Z 2 _ 4 W$d$ )&$$&$' -$ +3'-1.. ) J,:('.,$%, ), '&1-.(&$ b1&f1, 2 ^ ' (OSD) > T ETE #$ '-II &', -, I = t,t +T]2 -, f,'. %&$.($-, '-+ I IR n,. - J(1,:('., -$ I+&)-(. '%1(+, ), IR n $&.3.,. ''&%(3 K -$, $&+G,.,. -$, M&$%.(&$ l 1 (I ).,1' -, (t,y) I IR n, f(t,y),y l(t)(1 + y 2 ) 1&+' 1, I+&f1*G, ), F-%ie 4j )G,. - G &($' -$, '&1-.(&$ b1&f1, 4 4_

8 m j * +, #&(+ Z 2 _ 4 > B> E F&$'()3+&$' 1, I+&f1*G, ), F-%ie '-(/$. ^ y (t) = y 3 (t) + sin(t)y 2 (t) + 1 '-+ I =,+ Z )&$$3 )$' 4 4_ y() = y IR c,.i3&+*g, I+3%3),$. I,-. 'JII1( -,+ /,% 1, I+&)-(. '%1(+, -'-,1 ), IR4 W$,Y,. 2 &$ (t,y),+ IR f(t,y),y = y 4 + sin(t)y 3 + y y 4 + y 3 + y 1 + y,. )&$% (1,:('., -$, '&1-.(&$ b1&f1, '-+ I =,+ 4 * * %) " $ # ((! $" &#% (' %(! 4 $&.(&$ ) J-$(%(.3 ), 1 '&1-.(&$ )- I+&f1*G, ), F-%ie I,-. l.+, I+3%('3, ), 1 MH&$ c '-(/$., ^ N OPQCRCSQ T ETE #$ )(. -, 1, I+&f1*G, ), F-%ie 4j )G,. -$, '&1-.(&$,. -$, ',-1, 'J(1 )G,. -$, '&1-.(&$ b1&f1,,. '(.&-., '&1-.(&$ 1&%1,,'. 1 +,'.+(%.(&$ ), %,.., '&1-.(&$ b1&f1, 4 c,.i3&+*g, '-(/$.,'. -$.i3&+*g, M&$)G,$.1 ) J-$(%(.3 ), 1 '&1-.(&$ ^ ' (OSD) > TET E #$ '-II &', -, I,'. ), 1 M&+G, t,t + T] &- t,t + T &- t,+ 2 -, f,'. %&$.($-, '-+ I IR n,. - J(1,:('., -$, M&$%.(&$ l 1 (I ).,11, -, t I, y,z IR n, f(t,y) f(t,z),y z l(t) y z 2 4 _ 1&+' 1, I+&f1*G, ), F-%ie 4j )G,. -$, '&1-.(&$,. -$, ',-1, 4 * +, c J,:('.,$%,,'. f'3, '-+ 1,.i3&+*G, h &-+ 1 )3G &$'.+.(&$ ), 1J-$(%(.3 2 &$ +,$/&(, K Z 2 _ 4 c, i3&+*g, ), F-%ie\ c (I'%i(. iei &.i*',' I1-' M&+.,' ^ ' (OSD) > TET E,. - J(1,:('., -$ +3,1.,1 -, ^,'. -$, %&$'3 -,$%, )-.i3&+*g, I+3%3)$. I &-+ ),' 7@9?( =C A?(CR #$ '-II &', -, 1 M&$%.(&$ f,'. %&$.($-, '-+ I IR n (t,y)et(t,z) I IR n, f(t,y) f(t,z) y z 1&+' 1, I+&f1*G, ), F-%ie 4j )G,. -$, '&1-.(&$,. -$, ',-1, 4 4

9 t s u c JieI &.i*', Z +,/(,$. '(G I1,G,$. K %, -, '&(. \1(I'%i(. (,$$, I+ +II &+. K 4 4 f c y4 X &$$&$',$d$ -$ %&+&11(+, -(,'. -$, II1(%.(&$ )(+,%., ), %,.i3&+*g, 1 %&$%,+$, 1, %' 4 " ),' 3 -.(&$' )(Y3+,$.(,11,' 1($3(+,' 4 7SDSBB@CD> T ETEU 7@A 8>A O 9@RCSQA 8C OD>QRC>BB>A BCQO@CD>A ( 1 M&$%.(&$,'. 1($3(+, I+ +II &+. K y2 % J,'. K )(+, - J,11, ', G,. '&-' 1 M&+G, ^ f(t,y) = A(t)y(t) + b(t) 4 j,. -, 1JII1(%.(&$ t A(t),'. %&$.($-, '-+ I )$' 1J,$',Gf1, ),' II1(%.(&$' 1($3(+,' ), IR n 2 (IR n ),. -, b C (I ) 1&+' 1, I+&f1*G, ), F-%ie )G,. -$, '&1-.(&$,. -$, ',-1, 4 * +, I+,-/, ', '%($),,$ ),-: %' c ^ '(,'. %&G I%. &$ _4 I 2 f(t,y) f(t,z) = A(t)(y z) 4 m W$ I &'$. = maxt I A(t) (IR n ) -(,:('., I-(' -, I,'. %&G I%. 2 &$ &f.(,$. -, f,'. \1(I'%i(. (,$$, I+ +II &+. K c y4 ( $ J,'. I' %&G I%. I+,:,G I1, &$,$/('b, 1 '-(., ), 4 I 2 2 I = t,t + T2 tn = t + T 1/n4 " 1,:('., -$, '&1-.(&$ -$( -, b1&f1, '-+ t,t n ] -, 1J&$ $&., (t,t n ],y n )4 #$ I &', y(t) = limn + y n(t)4 F,.., 1(G (.,,:('., I-(' -, I &-+ t n t2 y n (t),'. %&$'.$., 4 c '&1-.(&$ (I,y),'. )&$% 1J-$( -, '&1-.(&$ )- I+&f1*G, 4 * * #" " $" '")!#)&!( #) " $ " X,++(*+, 1 -,'.(&$ ), 1J($ -,$%, ),' I,+.-+f.(&$' '-+ 1,' )&$$3,' '-+ -$, '&1-.(&$ ', %%i, 1 $&.(&$ ), '.f(1(.3 )- I+&f1*G, ), F-%ie 4 #$ '&-i(.,,$,y,. - J-$, I,+.-+f.(&$ '-+ 1,' )&$$3,' )- I+&f1*G, ), F-%ie 2 I1-' I+3%('3G,$. '-+ f,. '-+ 1 %&$)(.(&$ ($(.(1, y $ J,$.+ $, -, ),' /+(.(&$' '-+ 1 '&1-.(&$ y(t) -( I,-/,$. l.+, G.+('3,' 2,. 1,' I1-' I,.(.,' I &''(f1,' 4 $' 1 '.f(1(.3 )- I+&f1*G, %&$.($-, 2 (1 $ Je I' ) J,'I &(+ ), %&$'.+-(+, ),' G 3.i&),' $-G 3+( -,' Z)('%+*.,' -( '&(,$. '.f1,' 4 d$ ), I+3%(',+ %,.., $&.(&$ &$,$/('b, -$ I+&f1*G, ), F-%ie )(. I,+.-+f 3 2 ^.$. )&$$3' ^! -$ ($.,+/11, I IR2! -$, M&$%.(&$ f 2 )3d$(,,. %&$.($-, '-+ I IR n K /1,-+' )$' IR n ^! -$, M&$%.(&$ g2 )3d$(,,. %&$.($-, ), I )$' IR n! -$, 313G,$. α IR n,. ),-: +3,1' (ξ 1,ξ 2 ) IR 2

10 j +&-/,+ -$, M&$%.(&$ yε C 1 (I ).,11, -, y ε(t) = f(t,y ε (t)) + ξ g(t), t I, y IR n F&$)(.(&$ ($(.(1, y ε (t ) = y + ξ 1 α,t I, 4 c,.i3&+*g, '-(/$. I,+G,. ), G,'-+,+ 1J3%+.,$.+, -$, '&1-.(&$ )- I+&f1*G, ), F-%ie y,. -$, '&1-.(&$ )- I+&f1*G, ), F-%ie I,+.-+f 3 yε 4 ' (OSD) > TET E ( f,'. %&$.($-,,. /3+(d, ^ l 1 (I ), t I, y,z IR n, f(t,y) f(t,z),y z l(t) y z 2 4,. g,'. %&$.($-, 1&+' 1, I+&f1*G, ), F-%ie I,+.-+f 3 )G,. -$, '&1-.(&$ -$( -, yε,. /,% y(t) y ε (t) ξ 1 α exp((t)) + (t) = t t t exp ((t) (s)) ξ g(s) d t l(s)d 4 4 #$ +,G + -, )&$% -, 1 '.f(1(.3 ), 1J&I 3+.,-+,'. %&$.+ 13, I+ 1 M&$%.(&$ (t) -( )&(. +,'.,+ M(f1, I &-+,'I 3+,+ /&(+ -$, f &$$, '.f(1(.3 )- I+&f1*G, 4 * * # &)(! $" #!( - II,1&$' - J-$, M&$%.(&$ /,%.&+(,11,,'. )(., ), %1'', Ck '(,11, )G,. ),' )3+(/3,' I+.(,11,' %&$.($-,' ]-' - JK 1J&+)+, k4 #$ )&$$, 1,.i3&+*G, '-(/$. ' b-1+(.3 ),' '&1-.(&$' ^ ' (OSD) > TET E ( f : IR n+1 IR n,'. ), %1'', C k 1&+'.&-., '&1-.(&$ )- I+&f1*G, ), 2 F-%ie,'. ), %1'', C k+1 * +, I+,-/, ', M(. I+ +3%-++,$%, c ^! k = 2 f,'. %&$.($-, 4 h + iei &.i*', 2 y : I IR n,'. )3+(/f1, )&$% %&$.($-, 4 h + %&$'3 -,$. 2 y (t) = f(t,y(t)),'. %&$.($-,,. )&$% y,'. ), %1'', C 1 4! #$ '-II &', -, 1, +3'-1..,'. /+( K 1J&+)+, k 12 1&+' y,'. - G &($' ), %1'', C k 4 F&G G, f,'. ), %1'', C k 2 (1 'J,$'-(. -, y,'. ), %1'', C k,. )&$% -, y,'. ), %1'', C k+1 4

11 t s r 7@B?9B 8>A 8ODCkO>A A9??>AACk>A 8 9Q> ASB9RCSQ >Q SQ?RCSQ 8>, %1%-1 ),' )3+(/3,' y f c '-%%,''(/,' ), I+ +II &+. K ', M(. b+ %, -: )3+(/3,'.&.1,' ), I+ +II &+. K W$ )3+(/$. y t f t4 1 +,1.(&$ y (t) = f(t,y(t))2 &$ &f.(,$. ^ y (t) = f(t,y(t)) + y f(t,y(t)) y(t) t t = f(t,y(t)) + y f(t,y(t))y (t) t = f(t,y(t)) + y f(t,y(t))f(t,y) t #$ $&., ), G $(*+, %&$/,$.(&$$,11, 1 )3+(/3,.&.1, ), I+ +II &+. -.,G I' f ^ #$ )&$% f (1) (t,y) = f(t,y(t)) + y f(t,y(t))f(t,y) t y (t) = f (1) (t,y),. ) J-$, G $(*+, I1-' b3$3+1, &$ $&.,+ 1 )3+(/3, K 1J&+)+, k ^ y (k) (t) = f (k 1) (t,y) ( &$ )3+(/, -$, $&-/,11, M&(' %,.., ),+$(*+,,:I+,''(&$ 2 &$ &f.(,$. ^ y (k+1) (t) = f (k) (t,y) = f(k 1) (t,y(t)) + y f (k 1) (t,y(t))y (t) t = f(k 1) (t,y(t)) + y f (k 1) (t,y(t))f(t,y) t Z 4 4 Z 4 4j Z 4 4j _ Z 4 4j Z 4 4jj Z 4j Z 4 4j Z 4 4j Z 4 4j Z 4 4j c, %1%-1 ),' )3+(/3,'.&.1,' ), f I+ +II &+. K I,-/,$. )&$% ', M(+, b+ %, K 1 +3%-++,$%, t '-(/$., ^ f () Z (t,y) = f(t,y) 4 4j f(l 1) (t,y) f (l) (t,y) = t y (l+1) (t) = f (l) (t,y) * * )" " ( ( )&'%( #" + y f (l 1) Z m (t,y)f(t,y) 4 4 Z m 4 4 _ X $' %,.., M&+G, 1 1( , '-+ 1J3.-),.i3&+( -,,. 1 +3'&1-.(&$ $-G 3+( -, ),' 2,'. /'., &-+ $, %(.,+ -, -,1 -,' +3M3+,$%, Z Z 4 h 2 _ j _ 4 Z _ j Z _ 4

12 j 5 6 <! : ; 9 : ; < "< ; 9"!!! 9 " 6, I+&f1*G, ), F-%ie,'. I+.(%-1(*+,G,$. )I.3 K 1J3.-), ), 1J3/&1-.(&$,$.,G I' ) J-$ c 'e'.*g, ), )(G,$'(&$ d$(, K I+.(+ ) J-$ 3.. ($(.(1 )&$$3 #$,$.,$) I $,$',Gf1, 4 2 ), /+(f1,' ($)3I,$)$.,' I,+G,..$. ), )3%+(+, ), MH&$ -$(/& -, -$ 'e'.*g, 4 c, 'e'.*g,,'. )(. ), )(G,$'(&$ d$(, '( 1J3..,'. 1-( -''( ), )(G,$'(&$ d$(, 2 &` I1-' I+3%('3G,$. '( %,. 3.. II+.(,$. K -$,'I%, ), )(G,$'(&$ d$(, 4 #$ )3'(b$, 1, I1-' '&-/,$. %,' 'e'.*g,' '&-' 1,.,+G, ), 'e'.*g,' )e$g ( -,' &-,$%&+, Xe$G ( -, $&$ 1($3(+, 4 X,++(*+, %,.., )3'(b$.(&$.+*' b3$3+1, 2 ', %%i, -$, f+$%i, f(,$ I+3%(', ),' G.i3G.( -,' II1( -3,' -( %&$'('.,,$ 1J3.-), -1(..(/, ),' 'e'.*g,' )e$g ( -,' $&$ 1($3(+,' 2 1, I1-' '&-/,$. ), )(G,$'(&$ d$(, 4 W$,Y,. 2 '( 1, %' ),' 'e'.*g,' 1($3(+,','. '', '(G I1, 1,' 'e'.*g,' )e$g ( -,' $&$ 1($3(+,' &Y+,$. -$,.+*' b+$), +(%i,'', ), %&G I &+.,G,$. 2 4 c,' I+($%(I1,' 3.-),' ), %,' 'e'.*g,' '&$. 1(3,' K ),' -,'.(&$' ), '.f(1(.3 ), +,%i,+%i,' 2 ) J,$',Gf1, ($/+($.' Z3 -(1(f+, %e%1,' 1(G (.,',. ), I &($.' %+(.( -,' ($'( -, 1,' f(m-+%.(&$' 2 4 c,' $&.(&$' ), %i&' I+,$$,$. $(''$%, )$' 1J3.-), ),.,1' 'e'.*g,' 4 -+ %, '-],. &$ I,-. %(.,+ 2 ),' +3M3+,$%,' %1''( -,' %&G G, Z 2 _ j 2 _ j 2 _ Q S 8)B> DSC> DO8@R>9D E #$ I+3',$., )$' %,.., I+.(, -$,:,G I1, %1''( -, ), 'e'.*g, )e$g ( -, 4 " 1 ' Jb(. ) J-$ G &)*1, I+&(,! I+3).,-+ +3b(. I+ 1,' 3 -.(&$' '-(/$.,' ^ y(t) = y 1 (t) y 2 (t) ], y = f(t,y) = αy 1 βy 1 y 2 γy 2 + δy 1 y 2 c,' )(Y3+,$.'.,+G,' ), %,' 3 -.(&$' I,-/,$. l.+, ($.,+I+3.3 ), 1 MH&$ '-(/$., ^! y 1 (t),'. 1 I &I-1.(&$ ), I+&(, K 1J ($'.$. t2! y 2 (t),'. 1 I &I-1.(&$ ), I+3).,-+ K 1J ($'.$. t2 ] j 4_! α,'. 1,.-: ), $(''$%, G &($' 1,.-: ), )3%*' $.-+,1 ),' I+&(,' Z α > 2!,'. 1 I+&ff(1(.3 ), +,$%&$.+,,$.+, -$, I+&(,,. -$ I+3).,-+ %&$)-('$. K 1 )(G ($-.(&$ β ),' I+&(,' 2! γ,'. 1,.-: ), G &+.1(.3 $.-+,1 ),' I+3).,-+' '$' I+&(, Z γ < 2! δ,'. 1, M%.,-+ ), %+&(''$%, ), 1 I &I-1.(&$ b+ %, K 1 +,$%&$.+, ), I+&(,',. ), I+3).,-+' 4 X,' /1,-+'.eI( -,' I &-+ %,' %&,a %(,$.' I &-+ %,' %&,a %(,$.' I,-/,$. l.+, I+(',' ), 1 '&+., ^ α =.25,β =.1,γ = 1.,δ =.1 j 4 h +.$. ) J-$, )&$$3, ($(.(1, y()2 1,' -,'.(&$' -1(..(/,' -( I,-/,$. ', I &',+ %&$%,+$$. %, 'e'.*g, I,-/,$. l.+, W'. %, -, 1,' ),-: I &I-1.(&$' I,-/,$. /(/+, &- )('I+.+, &-,$%&+, ^ 2 ' J3 -(1(f+,+ W$ M(. 2 &$ I,-. &f',+/,+ -, %,.ei, ), 'e'.*g, I &''*), ),' '&1-.(&$' I 3+(&)( -,' I &-+ %,+.($,' /1,-+' ($(.(1,' +,:,G I1, I+.$. ), 4 h 2 y() = (8,3) T &$ +,.+&-/, %,.., G lg, 2

13 s t s os o o /1,-+ - f &-. ) J-$ %,+.($.,G I' T 4 #$ (11-'.+, %,.., '&1-.(&$ K 1 db-+, 4_ &- 1J&$.+%, 1,'.+],%.&(+,' y1 (t),. y 2 (t) ($'( -, 2 )$' 1, I1$ ), Ii', (y 1,y 2 )2 1.+],%.&(+, J,:('.,$%, ), 4 c %,' '&1-.(&$' I 3+(&)( -,' -: 'e'.*g,' )e$g ( -,' $&$ 1($3(+,' %&$'.(.-, -$, '-],. ) J3.-), K I+.,$.(*+, 4 Values of y y(1) y(2) Time t y(2) solution.txt using 2: y(1) 4_! = SA?CBB@R>9D QSQ BCQO@CD> 8>D SB F, 'e'.*g,,'. %+%.3+('3, I+ 1J3 -.(&$ '-(\ /$., 2 θ (µ θ 2 ) θ Z + θ = 4j 4j F, 'e'.*g, I,-. l.+, %&G I+3 K -$, &'%(11.,-+ 1($3(+, &` 1 /('%&'(.3 Z I,-. Z θ + λ θ θ = λ %i$b,+ ), '(b$, '-(/$. θ4 c J3 -.(&$ ), 1J&'%(11.,-+ $&$\1($3(+, ), #$ ),+ I &1 I,-. l.+, G (', '&-' 1 M&+G, ) J-$ 'e'.*g, )- I+,G (,+ &+)+,,$ I &'$. y = (y1,y 2 ) T = (θ, θ) ^ ] ] θ y y 2 Z m = f(t,y) = (µ θ 2 ) θ = 4j 4 θ (µ y1 2)y 2 y 1-31('&$' +I(),G,$. -$, 3.-), -1(..(/, ), %, 'e'.*g, 4 #$ I,-. +,G + -,+.&-. ) Jf &+) -, 1, I &($. (,),'. -$ I &($. d:, Z3 -(1(f+, )- 'e'.*g, #$ I,-.,$'-(., )3%(),+ ) J3.-)(,+ 1 4 '.f(1(.3 ), %, I &($. d:, &-+ %,1 &$ %1%-1, 1 G.+(%, %&f(,$$,,$ ), Z.f(1(.3 )- 4 h 2 (,) f 'e'.*g, 1($3+('3,$ (,) 2 '&(. ^ Jac(f)(,) = 1 1 µ ] j 4 c,' /1,-+' I+&I+,' ), %,.., G.+(%, '&$. 1,' '&1-.(&$' ), 1J3 -.(&$ λ 2 ), )('%+(G ($$. µλ+1 = = µ 2 #$ &f.(,$. )&$%.+&(' %' 44 )('.($%.'^ _4 µ < 22 1,' /1,-+' I+&I+,' '&$. )&$% %&G I1,:,' %&$]-b-3,' ), /1,-+' λ i = µ 2 ± i1 2 4 W:G ($&$' 1,-+ I+.(, +3,11, 2 I &-+ )3.,+G ($,+ ), 1 '.f(1(.3 ^ Z 2 < µ < 2 1, I &($. (,),'. -$ M&e,+..+%.(M

14 m Zf µ = 2 1, I &($. (,),'. -$ %,$.+, Z% < µ < 22 1, I &($.,'. -$ M&e,+ +3I-1'(M 4 µ = 22 &$ &f.(,$. -$, /1,-+ I+&I+, )&-f1, '-(/$. 1 /1,-+ ), µ Z I &-+ 1, I &($.,'. ($'.f1, λ = 1 µ = 2 Zf λ = 1 I &-+ µ = 2 1, I &($.,'. '.f1, j 4 µ > 22 1,' /1,-+' I+&I+,' '&$. +3,11,' ^ λ i = µ 2 ± 1 µ 2 2,. )&$% 4 Z 1, I &($.,'. ($'.f1, '( µ > 2 Zf 1, I &($.,'. '.f1, '( µ < 2 h &-+ %&$%1-+, 1, I &($.,'. '.f1,,...+%.(m I &-+,. ($'.f1, I &-+, 2 (,) µ < µ > 4 c %i$b,g,$. ), %&G I &+.,G,$. I &-+ -( $&-' M(. I'',+ ) J-$ M&e,+ '.f1, K -$ M&e,+ µ = ($'.f1, 'JII,11, -$, f(m-+%.(&$ X $' $&.+, %' I+3%(' (1 'Jb(. ) J-$, f(m-+%.(&$ ), &IM g &-' (11-'.+&$' %,' I+&I+(3.3' '-+ 1,' db-+,',. 4 4j Values of y Values of y y(1) y(2) Time t Time t 4! y(1) y(2) y(2) µ =.1 y(2) y(2) vs. y(1) y(1) y(2) vs. y(1) y(1) µ =. *

15 s t s os o o q Values of y Values of y Values of y y(1) y(2) Time t y(1) y(2) Time t y(1) y(2) Time t y(2).8.6 µ =.1 y(2) y(2) vs. y(1) y(1) 3 µ = 1 y(2) y(2) vs. y(1) y(1) µ = 5 5 y(2) vs. y(1) y(1) 4j! *

16 m #$ I,-. +,G + -,+ ), I1-' -, )$' 1, %' &` 1, I &($. (,),'. ($'.f1, 2 &$ &f',+/, -, 1.+],%.&(+,.,$) /,+' -$, '&1-.(&$ I 3+(&)( -, 4 F,.eI, ), '&1-.(&$ I 3+(&)( -,,'. II,13, %e%1,' 1(G (.,,$.i3&+(, ),' 'e'.*g,' )e$g ( -,' 4 c, %i$b,g,$. ), %&G I &+.,G,$. -1(..(M ), 1 '&1-.(&$ -, $&-' /&$' II,13 f(m-+%.(&$ ), &IM,'. b3$3+1,g,$. (11-'.+3,,$.+H$. 1 '&1-.(&$ )$' -$, %&-I, )- I1$ ), Ii', Z',%.(&$ ), &($%+3 I+,:,G I1, I &-+ h 2 θ = y2 = 4 F&$'()3+&$' G ($.,$$. -$ +3b(G, M&+%3 I+ -$, M&+%, '($-'& )1, (11-'.+3 '-+ 1 db-+, m 2 4 ^ θ (µ θ 2 ) θ + θ = acos(ωt) #$ %&$'.., 1&+' -, 1,' %e%1,' 1(G (.,' ), 1 '&1-.(&$ 'J%%+&%i,$. '-+ %,+.($,' M+3 -,$%,' 4 j 4 9>A OQOD@B>AE #$ I,-. +,G + -,+ -, G lg, /,% -$, b+$), +3b-1+(.3 )- %ig I ), /,%.,-+' f(y,t) &$ &f',+/, -$, b+$), )(/,+'(.3 ), %&G I &+.,G,$. )- K 1 $&$ 1($3+(.3 )- G &)*1, 4

17 s t s os o o Values of y Values of y Values of y y(2) vs. y(1) Time t a = 3, ω = 1 y(2) vs. y(1) Time t a = 3, ω = Time t a = 3, ω = 8 y(2) vs. y(1) Values of y Values of y Values of y 2 15 y(2) vs. y(1) Time t 1.5 a = 5, ω = 1 y(2) vs. y(1) Time t a = 3, ω = 4 y(2) vs. y(1) Time t a = 3, ω = 12! *

18 mm " ;9" ; " ;9 " F&G G, $&-' 1J/&$' )3] K '&-1(b$3 2 1 )&$$3, ) J-$, ',-1,,'. '&-/,$. ($'-a '$., I &-+ )3d$(+ -$ I+&f1*G, f(,$ I &'3 4 X $' 1, I+b+Ii, )&$$3, ) J-$, /1,-+ ($(.(1, %&$)-(. K 1 )3d$(.(&$ )- I+&f1*G, ), F-%ie 4 X $' %, I+b+Ii, 2 &$ f &+), -$ $&-/,-.ei, ), I+&f1*G, &- ),' %&$)(.(&$' '&$. )&$$3,' -: f &+)' ), 1J($.,+/11, #$ $&G G, %,' I+&f1*G,' I &- '&-/,$. )3'(b$3' I+ 1,-+ %+&$eg, $b1(' &-$)+e #1-, +&f1,g Z% & 2 h 4 & Z Z _ 4 _ * (!( N OPQCRCSQ T E EU #$ II,11, -$ 1, I+&f1*G, '-(/$. ^.$. )&$$3' ^! -$ ($.,+/11, I = t,t] IR2! -$, M&$%.(&$ f 2 )3d$(,,. %&$.($-, '-+ I IR n K /1,-+' )$' IR n ^ f : I IR n IR n _ (t,y) f(t,y)! -$, M&$%.(&$ g2 )3d$(,,. %&$.($-, '-+ IR n IR n K /1,-+' )$' IR n 2 +&-/,+ -$, M&$%.(&$ y C 1 (I ).,11, -, g : IR n IR n IR n j (u, v) g(u, v) y (t) = f(t,y(t)), t I, y IR n F&$)(.(&$ -: 1(G (.,' g(y(t ),y(t)) =, m #$ $&., 1,' G.+(%,' ] %&f(,$$,' ), g(u,v) I+ +II &+. - I+,G (,+,. - ',%&$) +b-g,$.' ), 1 MH&$ '-(/$., ^ R = g u, S = g Z m 4 4 v X $' 1 I+.( -, %,' %&$)(.(&$' -: 1(G (.,' '&$. 1, I1-' '&-/,$. 1($3(+,' &$ 1,' $&., 1&+' 2 2 ^ /,% R,S IR n n,. b IR n )&$$3' 4 * * )')(! Ry(t ) + Sy(T) = b J,:('.,$%,,. 1J-$(%(.3 ),' I+&f1*G,' -: 1(G (.,','. -$ '-],. )31(%.,. (1,'. )(a %(1, ) J&f.,$(+ c ),' +3'-1..' -''( '(G I1,' -, %,-: &f.,$-' I &-+ 1, I+&f1*G, ), F-%ie

19 t o o o o u > B> 8> B SA?CBB@R>9D E /$. ^ h &-+ (11-'.+,+ %,. 'I,%. %&$'()3+&$' 1, I+&f1*G, -: 1(G (.,' '-(\ 2 u = u u() = u, u(t) = u T F, I+&f1*G, I,-. ', +,G,..+, '&-' -$, M&+G, )- I+,G (,+ &+)+, 1($3(+, ^ /,% y = u u ], A = 1 1 y (t) = A(t)y(t) + q(t) ] Ry(t ) + Sy(T) = b, R = 1 ], S = 1 m 4 _ ], b = u u T ] ( 1J&$ %&$'()*+, -$, ($.,+/11,,T] /,% T π2 1&+' -$, -$( -, '&1-.(&$ )- I+&f1*G, -: 1(G (.,' I,-. l.+, %1%-13, ^ y(t) = u cos t + u t u cos T Z m sin t 4 4_ sin T ( I+ %&$.+, &$ %&$'()*+, ),-: %' '&$. 1&+' K,$/('b,+ &-+ &$ -$, 2 T = π2 4 h u = u T 2 ($d$(.3 ), '&1-.(&$',. I &-+ u u T 2 &$ I' ), '&1-.(&$ 4 #$ %&$'.., )&$% -, 1,' +3'-1..' ) J,:('.,$%,,. ) J-$(%(.3 )3I,$),$. M&+.,G,$. ),' %&$)(.(&$' -: 1(G (.,',. ), 1J($.,+/11, ) J3.-), 4 c, +3'-1.. '-(/$. / $&-' I,+G,..+, ) J3%1(+%(+ %,.., +,G + -, 4 Q DOA9BR@R 8 >CAR>Q?> >R 8 9QC?CRO E I+,. +II,1&$' 1 )3d$(.(&$ '-(/$., ^ F&$'()3+&$' -$ I+&f1*G, -: 1(G (.,' 1($3(+, )3d$( y (t) = A(t)y(t) + q(t) Ry() + Sy(T) = b _j _ m N OP QCRCSQ TE ET #$ II,11, 1 M&$%.(&$ Y (t) : IR IR n n '.('M('$. 1, I+&f1*G, K /1,-+' ($(.(1,' '-(/$. ^ &` In n,'. 1 G.+(%, (),$.(.3 ), IR n n 4 Y (t) = A(t)Y (t) _ Y () = I n n _ W$ -.(1('$. %,.., '&1-.(&$ M&$)G,$.1, 2 1 '&1-.(&$ b3$3+1, )- I+&f1*G, -: 1(G (.,' 'J3%+(. ^ y(t) = Y (t) c + t ] Y 1 (s)q(s)ds _

20 m &` 1, I+G *.+, c IR n )3I,$) ),' %&$)(.(&$' -: 1(G (.,' ), 1 MH&$ '-(/$., ^ RY () + S Y (T)] c = b S Y (T) T Y 1 (s)q(s)ds _ #$ &f.(,$. 1&+' -$ +3'-1.. ) J,:('.,$%,,. ) J-$(%(.3 ), f', I &-+ 1,' I+&f1*G,' -: 1(G (.,' 1($3(+,' ^ ' (OSD) > TE EU F&$'()3+&$' -$ I+&f1*G, -: 1(G (.,' 1($3(+, )3d$( I+ y (t) = A(t)y(t) + q(t) Ry() + Sy(T) = b _ /,% A(t) : IR IR n n,. q(t) : IR IR n %&$.($-,' 4 c, I+&f1*G, -: 1(G (.,' m )G,. -$, 4_ -$( -, '&1-.(&$ '(,. ',-1,G,$. '( 1 G.+(%, )3d$(, Q,'. ($/,+'(f1, '&1-.(&$ ' J,:I+(G, 1&+' 4 c ^ y(t) = Y (t) c + * +, X(+,%., Q = R + S Y (T) t /,% c = Q 1 b S Y (T) ] Y 1 (s)q(s)ds T ] Y 1 (s)q(s)ds ( 1J&$ +,/(,$. K 1J,:,G I1, ), 1J&'%(11.,-+ 2 &$ /3+(d, M%(1,G,$. -, ^ Y (t) = cos t sint sin t cos t ] _ j m,. )&$% ] ] cos T sin T 1 Z m Q = R + S = 4 4 sint cos T cos T sint F,.., G.+(%,,'. '($b-1(*+, '(,. ',-1,G,$. '( X $' 1, %' %&$.+(+, &$,:('.,$%, T = jπ,j IN4 2,. -$(%(.3 4 * '' (&!( T E E E@ 9@RCSQA 8>A S9RD>A AC B>A >Q RD@?RCSQ >CSQ =>A S9RD>A >Q RD@?RCSQ?S D>AACSQ #$,$.,$) I+ 1,.,+G, f+ 31'.( -, -$ G (1(,- %&$.($- '&1(), -$()(G,$'(&$$,11, d$( $, I &-/$. ', )3M&+G,+ -, 1 )(+,%.(&$ 1&$b(.-)($1,

21 t o o o o r,$.+%.(&$,.,$ %&G I+,''(&$ 4 F, G (1(,- %&$.($- ',+ $&.3 ω = {x, x l} IR,. 1, I &($. %&-+$. ), %, G (1(,- ',+ $&.3 x4 ( 1J&$ %&$'()*+, -$ ',%.(&$ δx2 ), 1 I &-.+, 2 1J31&$b.(&$ %,.., ',%.(&$ ',+ )&$$3, I+ ^ (x + δx + u(x + δx)) (x + u(x)) = δx + (u(x + δx) u(x)) '&(. -$, )3M&+G.(&$ +,1.(/, ), ^ u(x + δx) u(x) δx X J-$, G $(*+, I1-' b3$3+1, &$ )3d$(+ 1 )3M&+G.(&$ ), 1 f+ %&G G, 2 ^ ε(x) = u(x) x = du(x) dx F&G G, )$'.&-. G (1(,- %&$.($- 2 %,.., )3M&+G.(&$,'. +,1(3, -:,Y&+.' ($.,+$,' )$' 1, f+ 2 % J,'. K )(+, 1 %&$.+($., 2 σ(x) I+ -$, 1&( ), %&G I &+.,G,$. 4 Fi&('(''&$' -$, 1&( ), %&G I &+.,G,$. 31'.( -, 1($3(+, ^ σ(x) = s(x)ε(x), x Ω &` c(x),'. 1 +(),-+ 31'.( -, )- G.3+(-,$ x4 #$ I &'.-1, ), I1-' -, 1 %&$.+($., ($.,+$, 3 -(1(f+, 1,' M&+%,',:.,+$,' II1( -3,',$ x2 '&(. ^ σ(x + δx) σ(x) + x+δx x f(s)ds = j %, -( ), G $(*+, 3 -(/1,$., 2 /,% '-a 'G G,$. ), +3b-1+(.3 I,-. ', G,..+, '&-' 1 M&+G, 1&%1, ^ dσ(x) dx = f(x), x Ω j _ F,.., 3 -.(&$ I,-. l.+, /-, %&G G, -$ %&$'3 -,$%, )- I+($%(I, M&$)G,$.1, ), 1 )e$g ( -, 2 ), 1 %&$',+/.(&$ ), 1 -$.(.3 ), G &-/,G,$. 1,.&-.,$ '..( -, 4 W$ -.(1('$. 1 1&( ), %&G I &+.,G,$. Z m &$ &f.(,$. -$, ) J&+)+, d ( c(x) du(x) ) = f(x), x Ω dx dx j h &-+ )3.,+G ($,+ -$, '&1-.(&$ K %,.., 3 -.(&$ 2 (1 %&$/(,$. ), I+,'%+(+, u(x),. u (x) -:,:.+3G (.3' 4 h 1-'(,-+' %' I,-/,$.,$/('b,+ %&G G, I+,:,G I1, ^ _4 c f++,,'. d:3,,$ x =,. 1(f+,,$ x = l2 '&(. 4 c f++,,'. 3.(+3,,$ x =,.,$ x = l2 '&(. j u() =,σ(l) = c(l)ε(l) = c(l)u (l) = jj u() = a,u(l) = a j m

22 m F&G G, )$' 1, %' b3$3+1 I &-+ 1,' I+&f1*G,' -: 1(G (.,' 2 (1 M-. I+,$)+, b+), K 1 )&$$3, ),' %&$)(.(&$' -: 1(G (.,' 4 F,11,' %( I,-/,$.,$,Y,. G,$,+ K 1J,:(b,$%, ) J -$, ($d$(.3 ), '&1-.(&$ &- K 1 $&$,:('.,$%, 4 #$ +,G + -,+ -, $&-' '&G G,' )$' ) J-$, &` 1 /+(f1, ($)3I,$)$., $ J,'. I' 1,.,G I' 4 F,%(,'. b3$3+1,g,$. I &-+ 1,' I+&f1*G,' ), G 3%$( -, ),' G (1(,-: %&$.($-' -$()(G,$\ '(&$$,1' 4 > B> AS9A >A@QR>9D F&$'()3+&$' -$, f++, i&g &b*$, ), 1&$b-,-+ 1 '-'\ I,$)-,,. '&-G (', K 1 b+/(.3 4 c J3 -.(&$,'. )3I1%,G,$. /-. )&$% ^ c d2 u(x) Z m dx 2 = ρg 4 4j &`,'. 1 b+/(.3,. 1 G '', /&1-G ( -, ), 1 f++, '&1-.(&$ b3$3+1, ), %,.., 3 -.(&$ g ρ 4 c,'. -$, I+f &1, -, 1J&$ I,-.,:I+(G,+ I+ ^ u(x) = mg 2c x2 + ax + b W$ -.(1('$. 1,' %&$)(.(&$' -: 1(G (.,' u() =,u &$ &f.(,$. (1) = 2 u(x) = mg ) (x x2 c 2 '&(. -$, )3M&+G.(&$,. -$, %&$.+($., ), j j ε(x) = mg c (1 x), σ(x) = mg (1 x) Z m 4 4j #$ I,-. +,G + -,+ -, 1, )3I1%,G,$.,'. G :(G-G K 1J,:.+3G (.3 1(f+, 1 f++,,. 1, )3M&+\ G.(&$,. 1 %&$.+($., '&$. G :(G-G K 1J,$%'.+,G,$. 4 =>A S9RD>A OB@ARC 9>A >Q >CSQ F&$'()3+&$' G ($.,$$. -$ G (1(,- %&$.($- '&1(), -$(\ )(G,$'(&$$,11, d$( $, I &-/$. ', )3M&+G,+ -, 1 )(+,%.(&$.+$'/,+'1,,$,:(&$ 4 #$ $&., 1, )3I1%,G,$..+$'/,+'1 2 1 *%i, 2 I+ v(x)4 #$ I &'.-1, -, 1, G &G,$. ),,:(&$ m(x),'. I+&I &+.(&$$,1 K 1 %&-+f-+, θ(x) ), 1 I &-.+, ^ Z m θ(x) = 4 4j 1 + u 2 '&(. ^ m(x) = c(x)θ(x) = c(x)u (x) Z m m u 2 (x) ( 1J&$ '-II &', ), I1-' -, $&-',$ I,.(.,' I,+.-+f.(&$',. -, 1 I &-.+, $, 'J31&(b$, I'.+&I ), I &'(.(&$ ($(.(1, 1&+' &$ I,-. '-II &',+ 2 θ(x) u -( ] &-, 1, + 1, ), 1 )3M&+G.(&$ ), (x),:(&$ 4 u

23 t o o o o ( 1J&$ '-II &', ), I1-' -, 1, G &G,$. ),,:(&$ 3 -(1(f+, 1,' G &G,$.',:.,+$,',$b,$)+3 I+ -$, M&+%, /&1-G ( -, f(x)2 &$ &f.(,$. ^ ( ) dx 2 c(x) d2 u(x) dx 2 = f(x) d2 m _ g &-' /&$' )&$% &f.,$- -$, ) J&+)+, 4 I &-+ 1 -,11, (1,'. $3%,''(+, ), I+,'%+(+, ),-: %&$)(.(&$' -: 1(G (.,' K %i -,,:.+3G (.3' 4 B>CSQ 8 9Q> S9RD> AS9A B@ D@kCRO Q >>D?C?>A #$ %&$'()*+, -$, I &-.+, i&g &b*$, ), 1&$b-,-+ 1,$ II-( '-+ ',' ),-:,:.+3G (.3',. '&-G (', K 1 b+/(.3 4 F1%-1,+ 1 *%i, G :(G-G ), 1 I &-.+, TE E E 7SQRD B> S 8>A h,-. l.+, -$ I,.(.,:,G I1, (''-, )- %1%-1 ),' /+(.(&$' 4 \ +($%(I, ), G (1.&$ h \ F&$.+ 1, &I.(G 1, *!( $" )"" 9>A A9D B>A SQ?RCSQA 8> D>>Q &! /!" ( '$! &'! ' &' "/', 5! " 6' $ "! -! "!&&! %! " +! " &! 6"% '$ 5 "'! &!&! " #"( '!! '! " &! 6 "% %& ", - 1 " " ' +% " ' +!% #$& ""! " +! '! % ) " #"( '!! &! 6")% '$ 5 "'! ", > B> 89 A AR) > 8> S9RD>A Z N > B@ ASB9RCSQ SQ8@ SQ?RCSQA 8> D>>Q E c '&1-.(&$ M&$)G,$.1, Y (t) I+3',$.3, - m,'. I1-' I+.(%-1(*+,G,$. )I.3, K 1J3%+(.-+, ),' '&1-.(&$' ), I+&f1*G,' ), 4 4 F-%ie )- M(. )- %i&(: &-+ 1J3.-), ),' '&1-.(&$' ), I+&f1*G, K /1,-+' -: 2 Y () = In n 4 h j 1(G (.,' 2 &$ I+3M*+, ($.+&)-(+, -$, -.+, '&1-.(&$ M&$)G,$.1, ^ Φ(t) = Q 1 Y (t) m

24 -( '.('M(.,11, -''( -$, i&g &b*$, ^ Φ (t) = AΦ Rφ() + SΦ(T) = I n n m j mm X, 1 G lg, G $(*+, -, I &-+ 1 '&1-.(&$ M&$)G,$.1, Y (t)2 &$ I,-.,:I+(G,+ 1 '&1-.(&$ )- I+&f1*G, -: 1(G (.,' I+ ^ y(t) = Φ(t)b + T G(t, s)q(s) ds &`,'. 1 )&$$3, I+ G(t,s) ^ Φ(t)RΦ()Φ 1 (s), s t G(t,s) = Φ(t)SΦ(T)Φ 1 (s), s > t =>A SQ?RCSQA 8> D>>Q 9QC 8C >QACSQ>BB>A m m X J-$, G $(*+, I1-' b3$3+1, 2 &$ I,-. ),d$(+ -$,$',Gf1, ), M&$%.(&$' ), +,,$ ''&%(3,' K -$ &I 3+.,-+ )(Y3+,$.(,1 1($3(+, ) J&+)+, n2.,1 -, ^ = dn dt n + a n 1(t) dn 1 dt n a 1(t) d dt + a Z m m (t) 4 4 &` 1,' M&$%.(&$' ai (t),i =,...,n 1 '&$. %&$.($-,' '-+ 1J($.,+/11, I 4 V$, M&$%.(&$ ), +,,$,'. -$ $&e- ($.3b+1 I+.(%-1(,+ #$ +II,11, - J-$ $&e- ($.3b+1 4 K(t,s) ''&%(3 K f,. g,'. )&$$3 I+ 1 M&+G-1, ^ g(t) = f(s)k(t,s)ds *!%$" # )( #" & (" #$ +,$/&(, I &-+ %,1 1J&-/+b, '-(/$. ^ Z 42 _ 4 m

25 u t uq 5 6 ;! 9! 9 ; * (!( F&$'()3+&$' ), $&-/,- -$, )- I+,G (,+ &+)+, F(t,y,y ) =. 4_ g &-' /&$' /- -, '( 1 G.+(%, %&f(,$$, v (F(t,y,v)),'. +3b-1(*+, &$ I+1(. ) J 2 (G I1(%(.,,. 1J&$ I &-/(. ', +G,$,+ - G &($'.i3&+( -,G,$. K 1 M&+G, I1-' %1''( -, Z 2 2 4_4 4 1J($/,+', 1&+' -, 2 v (F(t,y,v)),'. I+.&-. '($b-1(*+, 2 &$ I+1, ) J #$ I,-. )3d$(+ ), 4 G $(*+,.+*' b3$3+1, -$, ), 1 MH&$ '-(/$., ^ N OP QCRCSQ TEEU V$, ) J&+)+, 1,'. -$, +,1.(&$ (G \ I1( -$. -$, M&$%.(&$ y(t) : I IR IR ($'( -, ' )3+(/3, I &-/$. ', G,..+, '&-' 1 M&+G, ^ &` 1 G.+(%, %&f(,$$, F(t,y,y ) =. v (F(t,y,v)),'. I+.&-. '($b-1(*+, 4 4 X $' %, %' 2 1, I+&f1*G,,:I+(G, $&$ ',-1,G,$. -$, +,1.(&$,$.+, 1,' )3+(/3,' ), y G (' -''( ),' %&$.+($.,' Z+,1.(&$' 1b3f+( -,' $, I &+.$. -, '-+ t,. y4 V$ %' ) J,'. I+.(%-1(*+,G,$. ($.3+,''$. 1&+' -, 1, 'e'.*g, I,-. I+,$)+, 1 M&+G, ) J-$,,:I1(%(., _4 -bg,$.3, ) J-$, 3 -.(&$ 1b3f+( -, 4 " 1 'Jb(. ) J-$, ',G (\,:I1(%(., -, 1J&$ )3d$(+ ), 1 MH&$ '-(/$., ^ N OP QCRCSQ TEET #$ II,11, -$, '%\ 1(+, ) J&+)+, 1 -$ 'e'.*g, ) J3 -.(&$ -( G,.,$ ],- -$, M&$%.(&$ y(t) : I IR IR ($'( -, ' )3+(/3, I+,G (*+, ), 1 M&+G, ^ y = f(t,y,z) = g(t,y,z) 4j 4 m V$, ',G (\,:I1(%(.,,'. )&$% %&$'.(.-3, ) J-$, 4j )3I,$)$. ) J-$ I+G *.+, '-II13G,$.(+, z,. ) J-$, 3 -.(&$ 1b3f+( -, )] &($., 4j -( %&$.+($. 1 '&1-.(&$ ), 1J 4 #$ I,-. +,G + -,+ -, 1 G.+(%, %&f(,$$, ), 1J 3 -(/1,$., ^,'. %1(+,G,$. '($b-1(*+, 4 ] v (F(t,y,v)) = I 4

26 )(Y3+,$%,,$.+, -$ (G I1(%(., ZG.+(%, ] %&f(,$$, +3b-1(*+,,. -$,'. (G I &+\ c.$., F&$'()3+&$' 1J,:,G I1, '(G I1, )&$$3 )$' Z & 4 2 _ ^ y = z = y t 4 F1(+,G,$. 2 1,' M&$%.(&$' y = t,. z = 1 '&$. '&1-.(&$' )- I+&f1*G,,. %,%( '$' G lg, 1 )&$$3, ), %&$)(.(&$' ($(.(1,' ( &$ (G I &', -$, %&$)(.(&$ ($(.(1, +f(.+(+,,11,,'. ($%&$'('.$., 4 y() = y 2 K G &($' -, y = 4 #$ %&$'.., )&$% -, 1,' %&$)(.(&$' ) J,:('.,$%, ), '&1-.(&$',. ), %&$'('.$%, ),' )&$$3,' ($(.(1,' '&$. %&G I1*.,G,$. )(Y3+,$.,' I+ +II &+. - I+&f1*G, ), F-%ie 4 SRCSQ 8 CQ8C?> - ),1 ), 1J&+)+,,. )- ),b+3 ) J-$, 2 -$, $&.(&$ M&$)G,$.1,,'. 1 $&.(&$ ) J($)(%, 4 N OPQCRCSQ T EE #$ II,11, 1, $&Gf+, G ($(G-G ), )3+(/.(&$ $,%,''(+, I &-+ +3'&-)+, ), G $(*+, -$( -, y,$ M&$%.(&$ ), y,. ), Z% J,'. K )(+, ), )3d$(+ -$, t I &-+ y 4 c 1( , '-+ 1, '-],.,'. /'., 4 g &-' )&$$&$' )$' 1 '-(., -,1 -,' %' I+.(%-1(,+' ) J,. -,1 -,',:,G I1,' ) JII1(%.(&$',$ +,$/&e$. -: &-/+b,' '-(/$.' I &-+ I1-' ), )3.(1' ^ Z 42 _ * * ( &()" N OPQCRCSQ T EE G,..+, '&-' 1 M&+G, ^ &` A IR n n $ J,'. I' ), +$b I1,($ 4 & 2 _ 2 _ 4 #$ II,11, -$, 1($3(+, K %&,a %(,$.' %&$'.$.' 2 -$, I &-/$. ', Ay (t) + By(t) = f(t) 4 #$ ''&%(, K -$, 1($3(+, 1, M('%,- ), G.+(%, ),d$( I+ 1J,$',Gf1, ),' G.+(%,' (A,B) 2 #$ )(. -, 1, M('%,-,'. +,b-1(,+ '( 1, I &1e$&G, %+%.3+('.( -, A λb,λ C4 2 pm (λ) = det(a λb) $, ' J$$-1, - J-$ $&Gf+, ), I &($.' d$(' ), C4 #$ I,-. G &$.+,+ -, 1, I+&f1*G, I,-. )G,..+, -$, '&1-.(&$ ',-1,G,$. '( 1, M('%,- ''&%(3,'. +3b-1(,+ 4 X $' %, %' 2 &$ I,-. +G,$,+ 1J K 1 ^ y 1 + Cy 1 = f 1 (t) Ny 2 + y 2 = f 2 (t) 4 &`,'. -$, G.+(%, $(1I &.,$., Z N N ν =,N i, < i < ν 4 c J($)(%, ), $(1I &.,$%, 2 ν 2 ), N,'. -''( 1J($)(%, )- M('%,- ), G.+(%, ($'( -, 1J($)(%, )(Y3+,$.(,1 ), 1J X- M(. -, 1 4

27 u t u G.+(%, N '&(. $(1I &.,$., I,+G,. ) J,:I+(G,+,:I1(%(.,G,$. 1 '&1-.(&$ ), y2 (t) ^ ( y 2 (t) = N d ) 1 dt + I ν 1 f 2 (t) = ( 1) i N i f (i) 2 (t) Z 4 4 N OP QCRCSQ TEE #$ II,11, -$, 1($3(+, %$&$( -, K %&,a %(,$.' %&$'.$.' -$ 'e'.*g, ), 1 M&+G, ^ Ny Z + y = g(t) 4 4_ &`,'. -$ f1&% ), &+)$ $(1I &.,$. ),.(11, N ν 4 #$ +II,11, - J-$ f1&% ), &+)$,'. -$, G.+(%, ), 1 M&+G, ^ a a a Z N = a 1... a 7@A 8>A?S >?C>QRA k@dc@ B>A E #$ I,-. %&$'()3+,+ ),' 'e'.*g,' 1($3(+,' )&$. 1,' G.+(%,' A,. B )3I,$),$. )-.,G I' 4 X $' %, %' &$ I,-.,$%&+, %&$'()3+,+ 1, M('%,- ), G.+(%,,. 2 -$, M&+G, %$&$( -, ), +&$,%,+ G (' 1,' %&$)(.(&$' ), '&1/f(1(.3,. 1 +,1.(&$ /,% 1J($)(%, $, '&$. I1-'.+(/(1,' 4 * " ( &()" )" $" " " ") F&G G, &$ 1J )(. I+3%3),G G,$. 2 (1,'..+*' )(a %(1, ), )&$$,+ ),' I+&I+(3.3' b3$3+1,' ),' '$' I+3%(',+ -$ I1-' 1,-+ M&+G, &-+ %,1 )&$$&$' 1 )3d$(.(&$ ) J-$, M&+G, -'-,11, 4 h 2 ) J 4 N OP QCRCSQ TEE #$ II,11, M&+G, ),,'',$f,+b ),.(11, ) J-$, '&-' 1 M&+G, r > 1 '-(/$., ^ y 1 = F 1(t,y 1,y 2,...,y r ) y 2 = F 2(t,y 1,y 2,...,y r 1 )... y i = F Z i(t,y i 1,...,y r 1 ) 4 4_... i= y r 1 = F r 1(t,y r 2,y r 1 ) = F r (t,y r 1 )

28 m &` 1 G.+(%, F r. F r 2... F 2 F 1,'. $&$ '($b-1(*+, 4 V$, x ), r 1,'',$f x r 2,+b ), x.(11, 1 x r ', )3d$(. %&G G, 1 ^ y 1 = f(t,y 1,y 2 ) = g(t,y 1,y 2 ) &`,'. ($/,+'(f1, #$ I+1, -''( ) ',G (\,:I1(%(., ) J($)(%, gy _j W$ II1( -$. 1 )3d$(.(&$ Z I &-+,. &$ &f.(,$. -$, ),.(11, 4 4_ r = 2 r = 32 2 ^ y 1 = f(t,y 1,y 2 ) Z 4_ = g(t,y 1 ) /,% ($/,+'(f1,,. -$, ),.(11, gy1 f y2 3 ^ y 1 = f(t,y 1,y 2,y 3 ) y 2 = k(t,y Z 1,y 2 ) 4 4_ = g(t,y 2 ) /,% ($/,+'(f1, gy2 k y1 f y3 4 h &-+ ($M&+G.(&$ 2 -$, G.+(%, ),,'',$f,+b,'. -$ G.+(%, 1 M&+G, '-(/$., ^ DS DCOROA E c J($)(%, ),' M&+G,' ),,'',$f,+b,'. 3b1 K 1,-+.(11, 4 4_ * "" ' " $ &'' (&!( # #" " #&!( $" &)& " & "!)&(!" "! " ()#(! "!)( #" 7CD?9CRA OB>?RDC 9>AE c,' II1(%.(&$' ),' '&$. $&Gf+,-',' V$, ),' I1-' (G I &+.$.,' 4,'. 1 '(G-1.(&$ ),' %(+%-(.' 31,%.+( -,' %&G I &+.$. ),' +3'('.$%,' ),' %I%(.3',. ),' f &f($,' 2 4 W$,Y,. 1 G (',,$ 3 -.(&$' ),' %(+%-(.' I+ 1,' 1&(' ), (+'%i&y %&G I &+., ),-: Ii',' 1J3%+(.-+, 2 ^ ), 1 1&( ),' G (11,' -( %&$)-(. K ),' 3 -.(&$' )(Y3+,$.(,11,',. 1 1&( ),' $&,-)' -( %&$)-(. ),' 3 -.(&$' 1b3f+( -,' 4 #$ &f.(,$. )&$% 2 I+ %,.., G 3.i&), 2 $.-+,11,,G,$. 2 -$ 4 c, I1-' '&-/,$. 2 I+ )3+(/.(&$ '-%%,''(/, 2 &$ I,-. ', +G,$,+ K -$, '.$)+) 4 F,.., G $(I-1.(&$ $ J,'. I' %&$.+, I+.&-] &-+' 3/(),$.,,. M(. I,+)+, ), $&Gf+,-',' I+&I+(3.3' '.+-%.-+,11,' - I+&f1*G, 4 F J,'. I &-+ -&( ), $&Gf+,-: 1b&+(.iG,' $-G 3+( -,' '&$. -] &-+) Ji-( )3/,1&II 3' I &-+ +3'&-)+, )(+,%.,G,$. 1,' 4

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