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\"Lg΀꺞X]OT'0B6kGRTu9ߺI#[-_5s`S=Rã`9dX[Th恃TVpet'D\YKdChRQDs/F_xrD%EUK@7_am7?PK!x0drs/downrev.xmlDJ1F"6ۂbצE>fYLdn}z=|;uqlˍPI-v' ,mxK4*K8h֩v1MBO>DhqrYQ\k-==8?w_~`߰]=~1l%4JjmsXn1 E9.fzPK-![Content_Types].xmlPK-!1_a ._rels/.relsPK-!k)drs/shapexml.xmlPK-!x0qdrs/downrev.xmlPKswK:2___PPT9  r* &33  U   ~`,?Rectangle 6u`  [ 1Chapter 1 Introduction to Differential Equations 2 2G 332   f/??Rectangle 7'g J1.1 Definitions and Terminology (S)@& "& 33 33& 33*"    t  ^ -??Text Box 8'xTLD___PPT9&  dDifferential Equation (DE): any equation containing derivation (page 2, definition 1.1) x: independent variable ꁊxe y(x): dependent variable axe ~ 2D 2q "33@#CFF '  3  d 0 s A ?5%8c8c    ?Object 9"Pj(  [  DA ?Object 32 ~   33___PPT10i.Ќ) +D=' = @B +"  'P (     bx??bq_Gr}_HrbMn@S 3" z PK![Content_Types].xmlJ0*miG]`Hm6 Xwt?. g#Wi3E).+7 >ON ޑ1ˋz "+R RhB.} /<^ITխ%rHK4uK~I0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!F";drs/shapexml.xmlVMo6 t-Kd: iq9Hc'z)-z( SzڟM_? "顇A!Mr8>y3rd ޷"G"NyzqfuJ᧟<)zeApo-*zF-XNQ0DȜ*Lyd\Qtv#)a_ΊcG$iܷ4Ǖ~û߼_߽߼yۻ7_ըjDbH`ũ.Oo%]![..IDɍV?b("teDt#]WckE"w/o$ºت=퇶SR=c>moI)'js0Ͳ&D\XS%izuΤ^==g<ȳۡx4t|[q~84:D@҈5jra?X*4K(fOj*ٸI-r'*9f۝BϷӱ;QMS6><*[]WE=^*&Iౣ4wioh<+fx H B Ve{c;v; GɝNj0)1Z=KD7[+tZ~ZH^PKhvA,UقU+&f][{`C!Yw{7_ZRW9C* ܅YI E-ZR,=%(zMML|/@:#X[j3)pWP5"-J3*ZB?]*4 Yf*ۖO*o胁鞓S95acpb%)b{ +bV1KN|v]㮱VnF|(/ j`8:8qdA9G0_uLí}*8R:w YrwUnuٷ^/k|9C)S7CqRrv)by9q&$';jÖvFCg YBi$eKt05^}qLBMzaᡅ\aXUwٷ80^.ffd & g@u]OFv+KfA!c)D:֌oդɑI-mgj/ kչFt )_W b i99)R-By %36v;YmF.w1},Z*hΠBE %)]RL |wÿPK!ȹdrs/downrev.xmlDN0HH  Ԇ ziK`ox٦Iyz,p}`[q& kn"DdcRp zyk  T`b )Ceb8u'-}->m+,{N;z5T}K^`vjh]m O " ~Xv/j N7z!Wܒi9PK-![Content_Types].xmlPK-!1_a ._rels/.relsPK-!F";)drs/shapexml.xmlPK-!ȹldrs/downrev.xmlPKowK:2___PPT9  r* &33  r  ^@x??Text Box 4  & Note: In the text book, f(x) is often simplified as f notations of differentiation , , , , & & & . Leibniz notation , , , , & & & . prime notation , , , , & & & . dot notation , , , , & & & . subscript notationxW 2 (  (]  ~   BA ?Object 5=B ~   BA ?Object 6p ~   BA ?Object 7G ~   BA ?Object 8E ~   BA ?Object 9 o B    DA ?Object 11p    DA ?Object 12-    DA ?Object 13M m p    DA ?Object 14     DA ?Object 15     DA ?Object 16-     DA ?Object 17z *     DA ?Object 18      DA ?Object 19      DA ?Object 20      DA ?Object 21 - ]   33___PPT10i.ՌP+D=' = @B +") 0(`$(  $ $ bx??bq_Gr}_HrbMn@S 3"} w PK![Content_Types].xmlJ0*miG]`Hm6 Xwt?. g#Wi3E).+7 >ON ޑ1ˋz "+R RhB.} /<^ITխ%rHK4uK~I0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!hdrs/shapexml.xmlVMo6 t-KlY:ر Ag3-QT):vX{졇@O=詧9tE)+ECBp͛7|tg2oEDYzq]fuJ᧟<)zeApo͕*zFYNQ0DȜ*LQHV2Eyh:NӔ[0ů&ũԣT4[Mp?o_{ǯoܾݛIj VB$KXDe'7~ƒ.gQqr/ՏJt=](a|Y%2#:&$An#] \"ֻ7a;Ml~ҞC)d1h6Է$^KU]B_Efc]"xYȴo `u=kvFGг;뷏[JzynjzP+O#)JH $-X&:k"KcmNdJ 3IhD"ƾ'&ܦ v^|k;n0:x~T㔳5,Vn"h/O$Ks7g4،Mj~$¶{F=Chv:h0w$F;I1X <^8եjMDB*^B9R"RT'sZ@P T[Lgͺ7+TE  oZj)SUrpEAs!o,Z~Y${KP<(271+ȇ%(`op0i^A՘Oh(.,jRDt wk`ЀdlS*=bV{{NN؄dW81Z,9MZU4&#)-uy#sgh |u3 ,HUBnd1`ݽVգϮ͝}t<_Xf"OZD ,KIC4hiඛ7 GNRou}sn.Idr]L̀Wb~1ѩPc3xh!xi5]-;LYIjP]SQ?qfPzT 豎5[5iardҲ'r6t°V֢c]y BzlhdKhӜ-ə)Hh<ͩ,ᰱrn3ZuȆchRQDs/F_x|J%EU},M lbRUm PK!ȹdrs/downrev.xmlDN0HH  Ԇ ziK`ox٦Iyz,p}`[q& kn"DdcRp zyk  T`b )Ceb8u'-}->m+,{N;z5T}K^`vjh]m O " ~Xv/j N7z!Wܒi9PK-![Content_Types].xmlPK-!1_a ._rels/.relsPK-!h)drs/shapexml.xmlPK-!ȹidrs/downrev.xmlPKlwK:2___PPT9  r* &33  f $ ^x??Text Box 4Z' d(2) Ordinary Differential Equation (ODE): differentiation with respect to one independent variable.e 2*"33e  n $ ^`x??Text Box 5 vX  l(3) Partial Differential Equation (PDE): differentiation with respect to two or more independent variables.m 2)""33m  ~ $ BA ?Object 6wV  ~ $ BA ?Object 7  ~ $ BA ?Object 8 be  ~ $ BA ?Object 9 E   33___PPT10i.ՌP+D=' = @B +"3 :2p((  ( ( b`x??bq_Gr}_HrbMn@S 3"~ x PK![Content_Types].xmlJ0*miG]`Hm6 Xwt?. g#Wi3E).+7 >ON ޑ1ˋz "+R RhB.} /<^ITխ%rHK4uK~I0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK! ʃdrs/shapexml.xmlVMo6 t-KdY:C șXD8Y,Kn=CQ mlPI=l iÙ7oU &L9a<I֋YR'4 +VYO?IٯJükTox Z퉒qBTa*Zd*\T䭶t[͸SbVI=/$ɒն޽7׏?߼O޽tVsP j5+D.NE|^5~g"%"De\n6J1B$.0RY<HS>uz58>[)cx~#1ֽV'k?RVhCKX8ŴRU+u\DY?5!ǚj, 'Iϳvwb{xlD#Fn;h^% *XJj/E IbjΚ`ȳD.b\ >qwkg'&ܶ ۡu{Eoӳ7] qt;iهGE.V!<'4/2$ɳm+,{N;z5T}K^`vjh]m O " ~Xv/j N7z!Wܒi9PK-![Content_Types].xmlPK-!1_a ._rels/.relsPK-! ʃ)drs/shapexml.xmlPK-!ȹjdrs/downrev.xmlPKmwK:2___PPT9  r* &33  S ( ^@x??Text Box 4, ](4) Order of a Differentiation Equation: the order of the highest derivative in the equation "^ 2)5^  ~ ( BA ?Object 5ec n xB ( P33??Line 7pGp ( ^x??Text Box 9 g w 7th order( 2    ( DA ?Object 10tW   ( `x??Text Box 11@ &  w 2nd order( 2    33___PPT10i.ՌP+D=' = @B +"x w ,_(  , , bx??bq_Gr}_HrbMn@S 3" z PK![Content_Types].xmlJ0*miG]`Hm6 Xwt?. g#Wi3E).+7 >ON ޑ1ˋz "+R RhB.} /<^ITխ%rHK4uK~I0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!'3drs/shapexml.xmlVo6@:lɒ:+AgZb-Qt(˰u@;0 i;?_#%H CA!M}{|d]*|`0$Id gU֓O?y\0Z*U[*^VD8R! 0RqE.*Vqf:)~>/ŇGdj[W~?~ǟ__o|M:V9m5"] g">?3ppT"JW_2NnpqCX\|!xOWJ_֩,ꈎI)AA>u °h^Hu/趱UImză}",V&Nz>T} }QvM⡦Kd%4lݝڞ3h Ig2OW:>/$a|?ZEKQT=EBҲra?X*,K8OjjٺI-r+*9vώa`{۽ mz]yfT \ "h$ d4Čzn$^'^gأ0ñx4i[?&%X'b$.k&Bc%-"z|IKSvGi)ݽ{05퐂,;=ț/-ZJKi_FgR+\H EZQ,?(zMmLb@:cXXj3+pWR52-J'Tr)~( [V5X0h@Je-xWg1T{==rf s2]c݌HhSpqA+)tp,ȒSˣa[TQj!su M05ųoso_XA9Oj~Fc;9SHo@nKR'%.mzN$w^>+97 $r*B|.k`1n?T_-xh!xE5] ,;LfgYIjP]SQ?qʳ+PfH 걎5[7iardrCbۙ:Fun0jW5cC#XJcxt"r-F?-u@4gPn'GTRTߔ.)&U%FozPK!ȹdrs/downrev.xmlDN0HH  Ԇ ziK`ox٦Iyz,p}`[q& kn"DdcRp zyk  T`b )Ceb8u'-}->m+,{N;z5T}K^`vjh]m O " ~Xv/j N7z!Wܒi9PK-![Content_Types].xmlPK-!1_a ._rels/.relsPK-!'3)drs/shapexml.xmlPK-!ȹldrs/downrev.xmlPKowK:2___PPT9  r* &33   , ^x??Text Box 4 $(5) Linear Differentiation Equation:% 2%%   , lx??Rectangle 7Mf] R   ~ , BA ?Object 6y G , ^ x??Text Box 8~ 1All the coefficient terms are independent of y. B2 2333333332   , ^`x??Text Box 9  5OProperty of linear differentiation equations: If and y3 = by1 + cy2, then / 2 <  d <:P   , DA ?Object 10x}08   , DA ?Object 11kOa+    , DA ?Object 12 ,m   33___PPT10i.ՌP+D=' = @B +"   0 (  0 0 bx??bq_Gr}_HrbMn@S 3" y PK![Content_Types].xmlJ0*miG]`Hm6 Xwt?. g#Wi3E).+7 >ON ޑ1ˋz "+R RhB.} /<^ITխ%rHK4uK~I0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!_צdrs/shapexml.xmlVMo6 t-Kl: iq9Hc'z)-z( SzڟM_? "顇A&Ej8>y3rd ޷"G"NyzqځEJEyL3YߺbO o-*zF-XNQ0DȜ*Lyd\QtN#)a_ΊcG$iܷ4Ǒ~û߼_߽߼yۻ7_ը?6Z.S]KB\$\>vI1*5zPK!ȹdrs/downrev.xmlDN0HH  Ԇ ziK`ox٦Iyz,p}`[q& kn"DdcRp zyk  T`b )Ceb8u'-}->m+,{N;z5T}K^`vjh]m O " ~Xv/j N7z!Wܒi9PK-![Content_Types].xmlPK-!1_a ._rels/.relsPK-!_צ)drs/shapexml.xmlPK-!ȹkdrs/downrev.xmlPKnwK:2___PPT9  r* &33   0 ^ x??Text Box 4Zi '(6) Non-Linear Differentiation Equation( 2((  ~ 0 BA ?Object 5 ,  ~ 0 BA ?Object 6Y@Y ~ 0 BA ?Object 7GP   33___PPT10i.ՌP+D=' = @B +" 4(  4 4 b??bq_Gr}_HrbMn@S 3" À PK![Content_Types].xmlJ0*miG]`Hm6 Xwt?. g#Wi3E).+7 >ON ޑ1ˋz "+R RhB.} /<^ITխ%rHK4uK~I0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!adrs/shapexml.xmlVMo6 t-Kd:C ș([D8Y,Kn=CQ mlPI$3o޼ᓧ<#L}=p,x$/֋Yhw,R*c +a^T5ed9-D8!s0F!Yɸ Y8FNSn'RIҸo5-i+woyo?7yOzswo$-Q6Z.KzoHx&j'7r[~/?1+%/DuD$ }麂_N58[+ax~#ֽV'i?BhC}KH8ŤTU+u\i=5!ǚ-Lˮw9}=?jÑJzenjzP+O#)JH $-X&:"KcmNdJ 3I.hD"ޭ۞"wr3hv l/|8qn($lã"}7"<'4S$``GTNLAvCʡXÊXU̒Ok5֪乁6\.i3eYR@m+,{N;z5T}K^`vjh]m O " ~Xv/j N7z!Wܒi9PK-![Content_Types].xmlPK-!1_a ._rels/.relsPK-!a)drs/shapexml.xmlPK-!ȹrdrs/downrev.xmlPKuwK:2___PPT9  r* &33  l 4 ^??Text Box 4  (7) Explicit Solution (page 6) The solution is expressed as y = f(x) (8) Implicit Solution (page 7) Example: , Solution: (implicit solution) or (explicit solution) W 2  +  33 333W  ~ 4 BA ?Object 5L  ~ 4 BA! ?Object 7gD  ~ 4 BA" ?Object 9 R   4 DA# ?Object 11 s   33___PPT10i.ՌP+D=' = @B +"  8(  8 8 b??bq_Gr}_HrbMn@S 5"~ x PK![Content_Types].xmlJ0*miG]`Hm6 Xwt?. g#Wi3E).+7 >ON ޑ1ˋz "+R RhB.} /<^ITխ%rHK4uK~I0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!u&drs/shapexml.xmlVMo6 t-Kd:C ș([D8Y,Kn=CQ mlPI=l iÙ7ou &T8a<q},;)1g}늕O?yRʂ0/{EZ*U2Z`k9UEd\Qtv#)a_LGʼn$iܷ4Ǖ~û߼_߽߼yۻ7_j F$KDDe'7~ƒ^"[..IDɍV?b(/D J :c1$!0mGWu[9ckE"w/o$ºت=XR=c>moI)'js0Ͳ W<Lˮw9}=?jÑJzenjzP+O#)JH $-X&:"KcmNdJ 3I.hD"ޭ۞"wr3hv l/|8qn($lã"}7!<'4S$m+,{N;z5T}K^`vjh]m O " ~Xv/j N7z!Wܒi9PK-![Content_Types].xmlPK-!1_a ._rels/.relsPK-!u&)drs/shapexml.xmlPK-!ȹjdrs/downrev.xmlPKmwK:2___PPT9  r* &33  B 8  x0e0e?Rectangle 4'gA  [ v"1.2 Initial Value Problem (IVP) "# #G 33#  8 ^@??Text Box 5'r  @A differentiation equation always has more than one solution. for , y = x, y = x+1 , y = x+2 & are all the solutions of the above differentiation equation. General form of the solution: y = x+ c, where c is any constant. The initial value (*g_(W x = 0) is helpful for obtain the unique solution. and y(0) = 2 y = x+2 and y(2) =3.5 y = x+1.5 @ 2 [ < 2f V da]3333333333 33n %*    f 80 s A$ ?5%8c8c    ?Object 11"$2(  [ 8 DA% ?Object 17*   B 8 X??Line 23: ] ` :  8 DA& ?Object 26   B  8 X??Line 28 ] `  33___PPT10i.ՌP+D=' = @B +"6 =5<(  < < b ??bq_Gr}_HrbMn@S 3" z PK![Content_Types].xmlJ0*miG]`Hm6 Xwt?. g#Wi3E).+7 >ON ޑ1ˋz "+R RhB.} /<^ITխ%rHK4uK~I0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!-drs/shapexml.xmlVMo6 t-KlY:CșXD;Y,Kn=CQ mlPI=l iÙ7ouK&T8a<q֋ZT4+VZO?I+ ü}kTk4hrZq%BTa*d%*\gt9MuSrVLEǗSIҸo5-i+woy7{/yswo&-Q6Z.]KB\$\Qwkg'&ܦ v^|k;n0:xvT Vn"h/O$Ks`4،Mj~$¶{Ǝ=Chv:x0w$V;I1X""^8եjMDB*^BsrEPgZ8Ziw47pݻS)c܃iҒ4Ve}.JR(jՒJf9/AhLDW2 <վ6á Q{U>+MOТt>K!ca5jAjm|Pt:>X996'f] r(ǰ"hg5kUft@ ԙB2,(?G.yU4ڧ1ȶc{7 wG)}{y6Η0D12Az3'U+7Xj,6/]8;GM{rG˵9mF+]%ĜKYXi.DS3շ_t*̯ZFu^e|}anbfF`RpTTolO\}n:^J!=ֱf\& vI1*5zPK!ȹdrs/downrev.xmlDN0HH  Ԇ ziK`ox٦Iyz,p}`[q& kn"DdcRp zyk  T`b )Ceb8u'-}->m+,{N;z5T}K^`vjh]m O " ~Xv/j N7z!Wܒi9PK-![Content_Types].xmlPK-!1_a ._rels/.relsPK-!-)drs/shapexml.xmlPK-!ȹldrs/downrev.xmlPKowK:2___PPT9  r* &33   < ^x??Text Box 6  The kth order differential equation usually requires k initial conditions (or k boundary conditions) to obtain the unique solution. solution: y = x2/2 + bx + c, b and c can be any constant y(1) = 2 and y(2) = 3 y(0) = 1 and y'(0) =5 y(0) = 1 and y'(3) =2 For the kth order differential equation, the initial conditions can be 0th ~ (k 1)th derivatives at some points. 2 e 233 33336@ =  ~ < BA' ?Object 7q N < ^x??Text Box 9Tc 4(boundary conditions (W N Tޞ)* 2*    P < `x??Text Box 10  4(boundary conditions (W N Tޞ)* 2*     < `@ ??Text Box 11 (initial conditions)* 2   33___PPT10i.ՌP+D=' = @B +" @(  @ @ b ??bq_Gr}_HrbMn@S 5"~ x PK![Content_Types].xmlJ0*miG]`Hm6 Xwt?. g#Wi3E).+7 >ON ޑ1ˋz "+R RhB.} /<^ITխ%rHK4uK~I0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!Q`3drs/shapexml.xmlVo6@:lɲ:-[]/=kH'E]zaaN; i9]b)G`vh i}{'"'LVC}XX$?ZO"gJQ\p6.Ye=9* 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~,Chapter 2 First Order Differential Equation - -G - " P f] ??Rectangle 3'g (2-1 Solution Curves without a Solution ) )& 33)   P ^`Y ??Text Box 4' Instead of using analytic methods, the DE can be solved by graphs (W).G 2C  *C      P ^ ] ??Text Box 5T !slopes and the field directions: " 2" "  ~ P BA0 ?Object 70 x < r  :׻[_a PC $ D} 17"'\ zB  P J?Line 8 /TC@/TzB  P J?Line 9z":z"9]a  P H@^ ?Text Box 10`AzQQX px-axis$ 2    |B  P L?Line 12.me@F3 D*  P HX ?Text Box 1324) D19J (x0, y0)\ 2""*"*"   b P H^ ?Text Box 1481me@׻[F the slope is f(x0, y0) 2 $$$$,$,$  2 P X`_ ?Oval 15-ɰC/E R   h P A1??âicon_by-nc-saPicture 16\\140.112.59.229\njns^S\njns^S\Hr k\Hr kICON{O\Creative CommensScp2.5\icon_by-nc-sa.tiff ]}_a  33___PPT10i.ՌP+D=' = @B +"H OG T/(  T T b`` ??bq_Gr}_HrbMn@S 3" y PK![Content_Types].xmlJ0*miG]`Hm6 Xwt?. g#Wi3E).+7 >ON ޑ1ˋz "+R RhB.} /<^ITխ%rHK4uK~I0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!drs/shapexml.xmlVMo6 t-Kd:ؖ Ag3-QT):qX{졇@O=詧9tE)#٠(z$3o޼ᓧ"'LVCs,x, 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Zill and Michael R. Cullen. N 6"*," {  fr @%9lES TC $ D} 10": t T DA2??Picture 7@VdESh T A1??âicon_by-nc-saPicture 16\\140.112.59.229\njns^S\njns^S\Hr k\Hr kICON{O\Creative CommensScp2.5\icon_by-nc-sa.tiffh;aJ%9lHN  33___PPT10i.ՌP+D=' = @B +" 0X(  X: X ^b ??Text Box 4N'  Example 2 dy/dx = sin(y), y(0) =  3/2 With initial conditions, one curve can be obtained q 2          8 q   X b` ??bq_Gr}_HrbMn@S 3" y PK![Content_Types].xmlJ0*miG]`Hm6 Xwt?. g#Wi3E).+7 >ON ޑ1ˋz "+R RhB.} /<^ITխ%rHK4uK~I0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!*kɨdrs/shapexml.xmlVMo6 t-Kd: iq9Hc'z)-z( SzڟM_? 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N~ 6"*," x  dp tf(LJ XC " D} 8"6 t X DA3??Picture 7t](LJh X A1??âicon_by-nc-saPicture 16\\140.112.59.229\njns^S\njns^S\Hr k\Hr kICON{O\Creative CommensScp2.5\icon_by-nc-sa.tiff['CfPG  33___PPT10i.ՌP+D=' = @B +"   @\ (  \ \ ba ??bq_Gr}_HrbMn@S 3" z PK![Content_Types].xmlJ0*miG]`Hm6 Xwt?. g#Wi3E).+7 >ON ޑ1ˋz "+R RhB.} /<^ITխ%rHK4uK~I0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!b?c.drs/shapexml.xmlVo6@:lɲ:+AgZb-Qt(˰u@;0 i;?_#%H CA!M}{|d]*|`0$Id,R) %'~W%a^ˁTZUdqBTa*O[d*\T䭶t[͸uS|^I=Ϗ$ɒն޽׏?_O޽tVsP j=kD.D|V5~g""Den.JB$𞮔0SY<HS>u z58>[+cx~#1ֽV'k?RV)hCKX8RU+u\DY?5!⊇j, iϳvwj{dbgw#7'xqwcg'&ܶڡu{Eoӳ7] It3YGE.V!<#4/2$ɳm+,{N;z5T}K^`vjh]m O " ~Xv/j N7z!Wܒi9PK-![Content_Types].xmlPK-!1_a ._rels/.relsPK-!b?c.)drs/shapexml.xmlPK-!ȹldrs/downrev.xmlPKowK:2___PPT9  r* &33   \ ^\ ??Text Box 2,  JAdvantage: It can solve some 1st order DEs that cannot be solved by mathematics. Disadvantage: It can only be used for the case of the 1st order DE. It requires a lot of timeT 2 $ (2 $P    33___PPT10i.ՌP+D=' = @B +"c jbP`J(  ` ` bf ??bq_Gr}_HrbMn@S 7"~ x PK![Content_Types].xmlJ0*miG]`Hm6 Xwt?. g#Wi3E).+7 >ON ޑ1ˋz "+R RhB.} /<^ITխ%rHK4uK~I0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!d2drs/shapexml.xmlVMo6 t-Kl: iq9eHc'z)-z( SzڟM_AQ$=9(Ig޼y'OYJ,{{XPD ݶH(h*8Y׬ɓ[y{B[e89X̨Tkd*\jMu SjJ= ON%IU޽׏?߾O޽V:m5"] '",*?3t\$\~#!=UV'j?\hC=KP8դPU+u\I>Mⱦ6`-eҳ^vθ=n{Wom~0VQc4W:>.(b|?ZYJQX"!iIj~ٰ,iis%SBlJrES' Q5%v=1ED=gPA^5m :-x0IهGEV=Ӭ7"h7K$M2%Fc(x H{͆5Ǝ=hC?t[-<w;cPzäkD,tHi4;uDB2^B99T"RD- CPT;KvTC  oZj)3rpEA !o,Z|Y$} P<(21+̆%(agp0i^NՄOp(/+RDt wk*aЀt[*=bV{{NFĄW8$1,ZE,>MZ4Ƃ:SZƑs%a캩[TiJ!su Mj?6ųoso_:[B9rzc;9SHo@lKRF%f#hSo#n@F]F-x{변3ppI"+,ė2`j zN>C ʬgq8a2 \L"LW΀꺞WܰT'0BVkRXu)߲I#ٶ350r+UcC#XLCxtd 'lEDFEDfIj8oAe ]FqѲKŝD6tLKE🖊:&3P|Q7ƣS*)c oJ`]7oPK!ȹdrs/downrev.xmlDN0HH  Ԇ ziK`ox٦Iyz,p}`[q& kn"DdcRp zyk  T`b )Ceb8u'-}->m+,{N;z5T}K^`vjh]m O " ~Xv/j N7z!Wܒi9PK-![Content_Types].xmlPK-!1_a ._rels/.relsPK-!d2)drs/shapexml.xmlPK-!ȹjdrs/downrev.xmlPKmwK:2___PPT9  r* &33  = `  xg 0e0e?Rectangle 2 h  [ qSection 2-6 A Numerical Method  G 33  R ` `@h ??Rectangle 3 Y Another way to solve the DE without analytic methods independent variable x x0, x1, x2, & & & & Find the solution of Since approximation    3 K"("*""*""*" "J" Z `0 s A4 ?5%8c8c    ?Object 30"j   [~B ` V??Line 4`   ` ``i ??Text Box 27=  sampling(S#j) 2 *     Z `0 s A5 ?5%8c8c    ?Object 34"G3k  [B  ` X??Line 37"   ` DA6 ?Object 38O    ` DA7 ?Object 39    ` `i ??Text Box 40J Y  f MRNޞvON ޑ1ˋz "+R RhB.} /<^ITխ%rHK4uK~I0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!Rdrs/shapexml.xmlVn6?%[,c_AAִDj$R`nv ̲.]uUsE)LPI]L iR{<]dL[c#o8 EJEyL3YߺbO xo-*zF-XNQ0DȜ*Lyd\QtN#)qRcIҸo5-i+woy7{/yswo&mQ j=kDs) NEtQ8KB .ID/] 1_}!bK% u"1$!ڝvn7p["ֻ7a;Ml8iO YgL<-"e⤗RUWmqYX6!!k)Ӿ2pIwlٙ؞3ۃpٝx4t|[q~84:D@҈5jqAQ?T*4iH(fOj*ٸĤa3N׷kہtm AoގjraQU vUZbdi;xc#EӬ1 l;ھn^kn8#'pr'$p2LJaP%-S]:vt-L$\ϯ%4;!GJZD ulA Sv'y.{hC1[v <؛/mJ+xn3w!EVQ˯T2dy ɸGSf&&re—HD@N[j3)6jjgE ڔNgTs)~$ ;V5\U4hBRU-x_g1R鞓S95acpb%)c{ +aAV1KN|v]VnF|(/ j`8:8ʂs%p|u3MTq*#su MAbg̽}| <]Zv"2!z3'U+7\jHI/x A-  $ۣAaE}^,a 4Z2Os% ŸbSf~O\Ϋoqa2 \"L6z7j'}n:^;B$zc͸~VM|2oy>v°v[-mLq0NӜ䈭ȉ)Hh<[PY2as.{sѪKD坅Q":YOKET(}1U7%KIUћ7PK!ȹdrs/downrev.xmlDN0HH  Ԇ ziK`ox٦Iyz,p}`[q& kn"DdcRp zyk  T`b )Ceb8u'-}->m+,{N;z5T}K^`vjh]m O " ~Xv/j N7z!Wܒi9PK-![Content_Types].xmlPK-!1_a ._rels/.relsPK-!R)drs/shapexml.xmlPK-!ȹhdrs/downrev.xmlPKkwK:2___PPT9  r* &33    d `m ??Rectangle 3,  |`Example: dy(x)/dx = 0.2xy y(xn+1) = y(xn) + 0.2xn y(xn )*(xn+1  xn). dy/dx = sin(x) y(xn+1) = y(xn) + sin(xn)*(xn+1  xn). .             ((    (  (   (  ((  (          ((    (  (  ((  (  ~B d V??Line 8- ~B d V??Line 9]  d `m ??Text Box 10  X_p dy/dx = sin(x), y(0) =  1, (a) xn+1  xn = 0.01, (b) xn+1  xn = 0.1, (c) xn+1  xn = 1, (d) xn+1  xn = 0.1, dy/dx = 10sin(10x) vOP[ ! 2  2 8      d bj ??Rectangle 11: TJ  qConstraint for obtaining accurate results: (1) small sampling interval (2) small variation of f(x, y) Jr ir   33___PPT10i.ՌP+D=' = @B +"    p hp (  h h F j ?Text Box 3[  c(a) 2"   h F m ?Text Box 4' d  c(b) 2"   h Fl ?Text Box 5K c(c) 2"   h F`l ?Text Box 6' d  c(d) 2"  p @nJT hC " D} 5" t h DA8??Picture 8@nJTh h A1??âicon_by-nc-saPicture 16\\140.112.59.229\njns^S\njns^S\Hr k\Hr kICON{O\Creative CommensScp2.5\icon_by-nc-sa.tiff;z7tyBN" h  h A1??âicon_by-nc-saPicture 16\\140.112.59.229\njns^S\njns^S\Hr k\Hr kICON{O\Creative CommensScp2.5\icon_by-nc-sa.tiffotzN" h  h A1??âicon_by-nc-saPicture 16\\140.112.59.229\njns^S\njns^S\Hr k\Hr kICON{O\Creative CommensScp2.5\icon_by-nc-sa.tiff3x7GwB J h  h A1??âicon_by-nc-saPicture 16\\140.112.59.229\njns^S\njns^S\Hr k\Hr kICON{O\Creative CommensScp2.5\icon_by-nc-sa.tiffoGz J  33___PPT10i.،@`+D=' = @B +" l(  l l bn ??bq_Gr}_HrbMn@S 3" { PK![Content_Types].xmlJ0*miG]`Hm6 Xwt?. g#Wi3E).+7 >ON ޑ1ˋz "+R RhB.} /<^ITխ%rHK4uK~I0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!@+drs/shapexml.xmlVo6@:dٖ:-[]/=kH'E]zaaN; i9]b)G`vh i}{'"'LVC}XX$?ZO";H(Oh.8Z|rPy5(RrjUz$JƱ YPmhhI+'=Ujs(&D\PS%zw`uL&( }Τ= ÉJe$Pb)*G(ZHZVC.u6KE%ڜvɔ sIi1&ލ"r={};=ۏ9q}݌jqAQx!<#4:(2$ɳm+,{N;z5T}K^`vjh]m O " ~Xv/j N7z!Wܒi9PK-![Content_Types].xmlPK-!1_a ._rels/.relsPK-!@+)drs/shapexml.xmlPK-!ȹmdrs/downrev.xmlPKpwK:2___PPT9  r* &33   l ^Q??Text Box 2ZwV X 2"  l ^Q??Text Box 3Z  Advantages -- It can solve some 1st order DEs that cannot be solved by mathematics. -- can be used for solving a complicated DE (not constrained for the 1st order case) -- suitable for computer simulation Disadvantages -- numerical error (xeON ޑ1ˋz "+R RhB.} /<^ITխ%rHK4uK~I0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!}drs/shapexml.xmlVo6@:dɲ:+AgZb-Qt(˰u@;0 i;?_#%H CA!M}{|d]*|`0$Idw-R) %'~W%a^ˁTZUdqBTa*O[d*\T-q:f:)~>/ŇGd,i+wo~ׯ{/yswo&m6Z.3UHzoHx*j'7r[~,. +%/TuD$Ҕ t麄_n58>[+c!~#1aN)+D%YL|V:., qCM5JfeMӮo^gjdbow"7 &xqwcg'&\wF^ώ:#?{ӵ7uOQ26?*r1zE~)&Iౣ4wih2+x H vàmcN'ƣNj0)1Z=+D7_&$t鴃ZH^XKh~ AE%-!(N VM^S[5{`jB!Yw{7_ZR&Cj ܥWZ}Y$+P}(2?=Lb@:cXXj3+pWR52-J'Tr)~( [V5X0h@Je-xWg1T{==rf s2]c݌HhSpqA+)tp,ȒSˣa[TQj!su M05ųoso_XA9Oj~Fc;9SHo@nKR'%sGEv;?v =z3W bM%_ˀ9X:jfWKZFuQg|WanbnF9cRpTllO\}n:9Y;Bzc͹Mxoyv6Q-mDq d! Ǣ"eeD,7ķb.{?+h%B\5^":QOKET(}U7%KIUћ7PK!ȹdrs/downrev.xmlDN0HH  Ԇ ziK`ox٦Iyz,p}`[q& kn"DdcRp zyk  T`b )Ceb8u'-}->m+,{N;z5T}K^`vjh]m O " ~Xv/j N7z!Wܒi9PK-![Content_Types].xmlPK-!1_a ._rels/.relsPK-!})drs/shapexml.xmlPK-!ȹldrs/downrev.xmlPKowK:2___PPT9  r* &33   p ^Q??Text Box 4'X  Exercises for Practicing (not homework, but are encouraged to practice) 1-1: 1, 13, 19, 23, 33 1-2: 3, 13, 21, 33 1-3: 2, 7, 28 2-1: 1, 13, 20, 25, 33 2-6: 1, 3     33___PPT10i.،@`+D=' = @B +" ƻ**t(  t t bQ??bq_Gr}_HrbMn@S 3" Å PK![Content_Types].xmlJ0*miG]`Hm6 Xwt?. g#Wi3E).+7 >ON ޑ1ˋz "+R RhB.} /<^ITխ%rHK4uK~I0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!~ !!drs/shapexml.xmlVϏ4#?XL&ij;6mʨv5g7q0wڙJ\,9p@H8!q2!4 zh~<98aĈ'[YdU~zGOnU"8̪nK)nUKZꄗZEA$LŢU ZQ&6N ç`]MˉPj"P#\߽{o}~z՛۷{%q9l  xv @jp3d AD?॥ji[ }3z\c٤x,OSzq]5r+pK7Ű{gŰn{<*jc)*GcBP K'WRQB]xc غW<Ԗze`#; ~:F'.h-T9@2@Ez^&pe~VE!E:br; %4^IMxTm.(%3)`^UM"$T^T u-m}0 9c6 AaX dtFӛwŵwS2fqM9}9Tf, [@X 뮛+' cM~9yYr6jzFnrOAzl׭\s y_"/mD92v2/h G b@*?+($ *"+e@9Ô/*r7Kcv-74o)RtoH"ueɄ)?P!p'PK!U0drs/downrev.xmlDN0DHH\uP$-|&:&8q47z;.&(:ؖ7 [%X.X0[iTp*рKSc8w=J6Ӣ[{zpTna5=י{wbxw JhJ_{>TLB@v˦PK-![Content_Types].xmlPK-!1_a ._rels/.relsPK-!~ !!)drs/shapexml.xmlPK-!U0xdrs/downrev.xmlPKzY:WF>___PPT9   *0 &33&33   Ȩe  Aw  tC ,gQ[HrbMn@S 33"    PK!8[Content_Types].xmlAN0EH%N@%邴K@`dOdlyLhoDX3'AL:*/@X*eRp208J妾)G,R}Q)=HiҺ0BL):T뢸WQDY;d]6O&8* VCLj"󃒝 yJ.;[wIC_ :{IOA !>Ø4 p;fɑ3׶Vc.ӵn(&poPK!8! _rels/.relsj0 }qN/k؊c[F232zQLZ%R6zPT]( LJ[ۑ̱j,Z˫fLV:*f"N.]m@= 7LuP[i?T;GI4Ew=}3b9`5YCƵkρؖ9#ۄo~e?zrPK!D? $Mdrs/e2oDoc.xml\[oG~( ( xދ5; U% )H\$TCO =3ٵWą\{ٹs;|;+.,,3/JbUJ^<4ɈMYBeyqUӅ/f,F*@*!\4j+U@)PV\JU[5 "НG㱘5fh\>Q)yL)籠bJL-/:Wt>\h{l4AJWsR!a\HgU-o5}szk gַvFTl9Mdst2~Y~!ob\Iͩ܈Q4<#'cYYd1xׁ@0ͱ:OÙ] %ǓDglX'K(4j_ޗVU8޾Adl gC& 8[D`~IْeZ-L q%4Xbzj@ džla$%gz N*7&@rUq3!\NWdз!*_K v%cÈ O891qYJu[cYYVdU!9ncD P_1 =7ss |GJ8891]}67]#oR#=1@C8zǮ꾆 PK!۫drs/downrev.xmlDA0!/"] U/=g6Y<3 \wֈ_9V0' s+.\λcR$UTS{ )*(CS)}^E?v5ql |D5r$siPbMے*n[ۗ> yfǃRn "PW{JrPK!)idrs/_rels/e2oDoc.xml.relsMK1 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