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" }{TEXT 301 0 "" }}{PARA 292 "" 0 "" {TEXT 301 0 "" }}{PARA 292 "" 0 "" {TEXT 301 328 "The ordinary differential equation s solved in Euler's and 2nd and 4th order Runge-Kutta method files wer e all of the first order. These methods can be used to solve higher or der ordinary differential equations by converting them into a coupled \+ system of first order equations. The following sheet demonstrates how \+ this is done. " }{TEXT 301 0 "" }}}{SECT 0 {PARA 293 "" 0 "" {TEXT -1 13 "Intialization" }{TEXT -1 0 "" }}{EXCHG {PARA 294 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }{MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 291 "" 0 "" {TEXT 300 16 "Section 1: Input" }{TEXT 300 0 "" }}{PARA 295 "" 0 "" {TEXT 302 78 "This differential equation used here to illustrate th is method is of the form " }{XPPEDIT 2 0 "A1*(diff(diff(y(x), x), x))+ A2*(diff(y(x), x))+A3*y(x)+A4 = 0" "6#/,**&I#A1G6\"\"\"\"-I%diffGI*pro tectedGF+6$-F*6$-I\"yGF'6#I\"xGF'F2F2F(F(*&I#A2GF'F(F-F(F(*&I#A3GF'F(F /F(F(I#A4GF'F(\"\"!" }{TEXT 303 1 "." }{TEXT 302 33 " The user may spe cify values for " }{TEXT 303 10 "A1, A2, A3" }{TEXT 302 28 ", which ma y be functions of " }{TEXT 303 1 "x" }{TEXT 302 5 " and " }{TEXT 303 3 "y, " }{TEXT 302 3 "and" }{TEXT 302 1 " " }{TEXT 303 2 "A4" }{TEXT 302 33 " which may only be a function of " }{TEXT 303 1 "x" }{TEXT 302 1 "." }{TEXT 302 0 "" }}{EXCHG {PARA 296 "> " 0 "" {MPLTEXT 1 304 13 "A1:=(x,y)->2;" }{MPLTEXT 1 304 0 "" }{MPLTEXT 1 304 5 "\nA2:=" } {MPLTEXT 1 304 7 "(x,y)->" }{MPLTEXT 1 304 1 "4" }{MPLTEXT 1 304 1 ";" }{MPLTEXT 1 304 0 "" }{MPLTEXT 1 304 5 "\nA3:=" }{MPLTEXT 1 304 7 "(x ,y)->" }{MPLTEXT 1 304 1 "6" }{MPLTEXT 1 304 1 ";" }{MPLTEXT 1 304 0 " " }{MPLTEXT 1 304 5 "\nA4:=" }{MPLTEXT 1 304 3 "x->" }{MPLTEXT 1 304 1 "7" }{MPLTEXT 1 304 1 ";" }{MPLTEXT 1 304 0 "" }}{PARA 297 "" 1 "" {XPPMATH 20 "6#>I#A1G6\"f*6$I\"xGF%I\"yGF%F%6$I)operatorGF%I&arrowGF%F %\"\"#F%F%F%" }{TEXT 20 0 "" }}{PARA 297 "" 1 "" {XPPMATH 20 "6#>I#A2G 6\"f*6$I\"xGF%I\"yGF%F%6$I)operatorGF%I&arrowGF%F%\"\"%F%F%F%" }{TEXT 20 0 "" }}{PARA 297 "" 1 "" {XPPMATH 20 "6#>I#A3G6\"f*6$I\"xGF%I\"yGF% F%6$I)operatorGF%I&arrowGF%F%\"\"'F%F%F%" }{TEXT 20 0 "" }}{PARA 297 " " 1 "" {XPPMATH 20 "6#>I#A4G6\"f*6#I\"xGF%F%6$I)operatorGF%I&arrowGF%F %\"\"(F%F%F%" }{TEXT 20 0 "" }}}{EXCHG {PARA 298 "" 0 "" {TEXT 305 41 "The boundary condition must be specified." }{TEXT 305 0 "" }}}{EXCHG {PARA 296 "> " 0 "" {MPLTEXT 1 304 6 "x0:=0;" }{MPLTEXT 1 304 0 "" } {MPLTEXT 1 304 8 "\ny0:=10;" }{MPLTEXT 1 304 0 "" }}{PARA 297 "" 1 "" {XPPMATH 20 "6#>I#x0G6\"\"\"!" }{TEXT 20 0 "" }}{PARA 297 "" 1 "" {XPPMATH 20 "6#>I#y0G6\"\"#5" }{TEXT 20 0 "" }}}{EXCHG {PARA 295 "" 0 "" {TEXT 302 15 "The derivative " }{XPPEDIT 2 0 "diff(y(x), x)" "6#-I% diffGI*protectedGF%6$-I\"yG6\"6#I\"xGF)F+" }{TEXT 302 4 " at " }{TEXT 303 6 "x = x0" }{TEXT 302 1 "." }{TEXT 302 0 "" }}}{EXCHG {PARA 296 "> " 0 "" {MPLTEXT 1 304 10 "dydx0:=10;" }{MPLTEXT 1 304 0 "" }}{PARA 297 "" 1 "" {XPPMATH 20 "6#>I&dydx0G6\"\"#5" }{TEXT 20 0 "" }}}{EXCHG {PARA 295 "" 0 "" {TEXT 302 9 "Value of " }{TEXT 303 1 "x" }{TEXT 302 10 " at which " }{TEXT 303 1 "y" }{TEXT 302 11 " is desired" }{TEXT 302 0 "" }}}{EXCHG {PARA 296 "> " 0 "" {MPLTEXT 1 304 6 "xf:=1;" } {MPLTEXT 1 304 0 "" }}{PARA 297 "" 1 "" {XPPMATH 20 "6#>I#xfG6\"\"\"\" " }{TEXT 20 0 "" }}}}{SECT 0 {PARA 291 "" 0 "" {TEXT 300 25 "Section 2 : Exact Solution" }{TEXT 300 0 "" }}{PARA 292 "" 0 "" {TEXT 301 81 "Th e following uses Maple's built-in tools to find the exact solution for the ODE." }{TEXT 301 0 "" }}{EXCHG {PARA 296 "> " 0 "" {MPLTEXT 1 304 28 "ODE:=A1(x,y)*diff(y(x),x,x)+" }{MPLTEXT 1 304 39 "A2(x,y)*diff (y(x),x)+A3(x,y)*y(x)+A4(x)" }{MPLTEXT 1 304 3 "=0;" }{MPLTEXT 1 304 0 "" }}{PARA 297 "" 1 "" {XPPMATH 20 "6#>I$ODEG6\"/,*-I%diffGI*protect edGF*6$-I\"yGF%6#I\"xGF%-I\"$GF*6$F/\"\"#F3-F)6$F,F/\"\"%F,\"\"'\"\"( \"\"\"\"\"!" }{TEXT 20 0 "" }}}{EXCHG {PARA 299 "> " 0 "" {MPLTEXT 1 306 46 "soln:=(dsolve(\{ODE,y(x0)=y0,D(y)(x0)=dydx0\})):" }{MPLTEXT 1 306 0 "" }}}{EXCHG {PARA 299 "> " 0 "" {MPLTEXT 1 306 19 "assign(soln) ; y(x);" }{MPLTEXT 1 306 0 "" }}{PARA 297 "" 1 "" {XPPMATH 20 "6#,(*(- I$expG6$I*protectedGF(I(_syslibG6\"6#,$I\"xGF*!\"\"\"\"\"-I$sinGF'6#*& \"\"##F/F4F-F/F/F4F5#\"$F\"\"#7*&F%F/-I$cosGF'F2F/#\"#n\"\"'#!\"(F>F/" }{TEXT 20 0 "" }}}{EXCHG {PARA 299 "> " 0 "" {MPLTEXT 1 306 102 "plot (y(x),x=x0..xf,title=\"Exact Solution Plotted on Given Interval\",lege nd=\"Exact y(x)\" ,thickness=2);" }{MPLTEXT 1 306 0 "" }}{PARA 300 "" 1 "" {GLPLOT2D 662 372 372 {PLOTDATA 2 "6'-%'CURVESG6%7S7$$\"\"!F)$\"# 5F)7$$\"3emmm;arz@!#>$\"3N%\\*f9&R0-\"!#;7$$\"3[LL$e9ui2%F/$\"3KyQSZF4\"F27$$\"3]mm\"zR'ok; FE$\"3_pE>KfD)4\"F27$$\"3w***\\i5`h(=FE$\"3Pi#)H;3#>5\"F27$$\"3WLLL3En $4#FE$\"3![!zacI'Q5\"F27$$\"3qmm;/RE&G#FE$\"3I%4rGs)4/6F27$$\"3\")**** *\\K]4]#FE$\"3tQ\"f(4!yF5\"F27$$\"3$******\\PAvr#FE$\"3I6#3UIK)*4\"F27 $$\"3)******\\nHi#HFE$\"3Yl/bFg_&4\"F27$$\"3jmm\"z*ev:JFE$\"3aW.KwBU!4 \"F27$$\"3?LLL347TLFE$\"3]JNUMM%H3\"F27$$\"3,LLLLY.KNFE$\"3W5%3*QWYv5F 27$$\"3w***\\7o7Tv$FE$\"3WdjA+8^l5F27$$\"3'GLLLQ*o]RFE$\"3?f/Z0Bjb5F27 $$\"3A++D\"=lj;%FE$\"3jc:Iv/qV5F27$$\"31++vV&RY2aFE$\"3#*G&eqKx%f&*Fas7$$\"39mm;zXu9cFE$\"3+)G)p4u@ 'Q*Fas7$$\"3l******\\y))GeFE$\"3Y^/f_fi+#*Fas7$$\"3'*)***\\i_QQgFE$\"3 xWPMOz28!*Fas7$$\"3@***\\7y%3TiFE$\"36Ga-_GWE))Fas7$$\"35****\\P![hY'F E$\"3.)QA&)\\!y8')Fas7$$\"3kKLL$Qx$omFE$\"3.jB9!Rn#=%)Fas7$$\"3!)***** \\P+V)oFE$\"3bfR8m\"f`?)Fas7$$\"3?mm\"zpe*zqFE$\"3q!\\5y18\"4!)Fas7$$ \"3%)*****\\#\\'QH(FE$\"3;(*f`e%p8z(Fas7$$\"3GKLe9S8&\\(FE$\"3%GB'3(f= Qe(Fas7$$\"3R***\\i?=bq(FE$\"37$yEiT'[ktFas7$$\"3\"HLL$3s?6zFE$\"3i4&* H7-1[rFas7$$\"3a***\\7`Wl7)FE$\"3\\cYdknv>pFas7$$\"3#pmmm'*RRL)FE$\"3S ^]c#[f&)p'Fas7$$\"3Qmm;a<.Y&)FE$\"3%\\sW54q8Z'Fas7$$\"3=LLe9tOc()FE$\" 3#Rv`R(QTXiFas7$$\"3u******\\Qk\\*)FE$\"3X)QdeG!\\PgFas7$$\"3CLL$3dg6< *FE$\"3a#Qycbd\"*z&Fas7$$\"3ImmmmxGp$*FE$\"3[Ls0UMB'e&Fas7$$\"3A++D\"o K0e*FE$\"3E)Hmn)=uf`Fas7$$\"3A++v=5s#y*FE$\"3c$*3Dp3sV^Fas7$$\"\"\"F)$ \"3M/RxqAn7\\Fas-%&COLORG6&%$RGBG$F+!\"\"$F)F_[lF`[l-%'LEGENDG6#Q+Exac t~y(x)6\"-%&TITLEG6#QIExact~Solution~Plotted~on~Given~IntervalFe[l-%+A XESLABELSG6$Q\"xFe[lQ!Fe[l-%*THICKNESSG6#\"\"#-%%VIEWG6$;F`[lF^[l;$\"1 k^hrf5!z%!#:$\"2'opy_`N;6Fj\\l" 1 2 2 0 10 2 2 6 1 4 2 1.0 45.0 45.0 1 1 "Exact y(x)" }}{TEXT 307 0 "" }}}}{SECT 0 {PARA 293 "" 0 "" {TEXT -1 31 "Section 3: Approximate Solution" }{TEXT -1 0 "" }}{PARA 301 "" 0 "" {TEXT 302 215 "We are going to use Runge-Kutta's 4th order method of solving ODE's, although any method could be used. Our first task i s to make the single second order ODE, a system of two first order ODE 's. Thus, the derivative " }{XPPEDIT 2 0 "diff(y(x), x)" "6#-I%diffGI* protectedGF%6$-I\"yG6\"6#I\"xGF)F+" }{TEXT 302 34 " is set equal to an other variable." }{TEXT 302 0 "" }}{EXCHG {PARA 295 "" 0 "" {XPPEDIT 2 0 "diff(y(x), x) = z" "6#/-I%diffGI*protectedGF&6$-I\"yG6\"6#I\"xGF* F,I\"zGF*" }{TEXT 302 28 ". This will be function #1, " }{TEXT 303 2 " f1" }{TEXT 302 26 " and used for determining " }{TEXT 303 1 "k" } {TEXT 302 16 " values for the " }{TEXT 303 1 "y" }{TEXT 302 15 " appro ximation." }{TEXT 302 0 "" }}}{EXCHG {PARA 296 "> " 0 "" {MPLTEXT 1 304 15 "f1:=(x,y,z)->z;" }{MPLTEXT 1 304 0 "" }}{PARA 297 "" 1 "" {XPPMATH 20 "6#>I#f1G6\"f*6%I\"xGF%I\"yGF%I\"zGF%F%6$I)operatorGF%I&ar rowGF%F%9&F%F%F%" }{TEXT 20 0 "" }}}{EXCHG {PARA 295 "" 0 "" {TEXT 303 32 "dz/dx = -(A2*z + A3*y + A4) / A1" }{TEXT 302 28 ". This will b e function #2, " }{TEXT 303 2 "f2" }{TEXT 302 26 " and used for determ ining " }{TEXT 303 1 "k" }{TEXT 302 16 " values for the " }{TEXT 303 1 "z" }{TEXT 302 15 " approximation." }{TEXT 302 0 "" }}}{EXCHG {PARA 296 "> " 0 "" {MPLTEXT 1 304 27 "f2:=(x,y,z)->-1/A1(x,y)*(A2" } {MPLTEXT 1 304 5 "(x,y)" }{MPLTEXT 1 304 5 "*z+A3" }{MPLTEXT 1 304 5 " (x,y)" }{MPLTEXT 1 304 10 "*y+A4(x));" }{MPLTEXT 1 304 0 "" }}{PARA 297 "" 1 "" {XPPMATH 20 "6#>I#f2G6\"f*6%I\"xGF%I\"yGF%I\"zGF%F%6$I)ope ratorGF%I&arrowGF%F%,$*&-I#A1GF%6$9$9%!\"\",(*&-I#A2GF%F2\"\"\"9&F:F:* &-I#A3GF%F2F:F4F:F:-I#A4GF%6#F3F:F:F5F%F%F%" }{TEXT 20 0 "" }}}{EXCHG {PARA 298 "" 0 "" {TEXT 305 48 "The step size will be used throughout \+ this sheet" }{TEXT 305 0 "" }}}{EXCHG {PARA 296 "> " 0 "" {MPLTEXT 1 304 15 "h:=(xf-x0)/5.0;" }{MPLTEXT 1 304 0 "" }}{PARA 297 "" 1 "" {XPPMATH 20 "6#>I\"hG6\"$\"+++++?!#5" }{TEXT 20 0 "" }}}{EXCHG {PARA 298 "" 0 "" {TEXT 305 96 "The following shows five steps of the 4th or der Runge-Kutta method for solving a system of ODEs." }{TEXT 305 0 "" }}}{SECT 0 {PARA 302 "" 0 "" {TEXT 308 6 "Step 1" }{TEXT 308 0 "" }} {PARA 303 "" 0 "" {TEXT 301 95 "Using the values of the initial condit ions, we find the value of the solution one step forward." }{TEXT 301 0 "" }}{EXCHG {PARA 296 "> " 0 "" {MPLTEXT 1 304 32 "X[0]:=x0; Y[0]:=y 0; Z[0]:=dydx0;" }{MPLTEXT 1 304 0 "" }}{PARA 297 "" 1 "" {XPPMATH 20 "6#>&I\"XG6\"6#\"\"!F(" }{TEXT 20 0 "" }}{PARA 297 "" 1 "" {XPPMATH 20 "6#>&I\"YG6\"6#\"\"!\"#5" }{TEXT 20 0 "" }}{PARA 297 "" 1 "" {XPPMATH 20 "6#>&I\"ZG6\"6#\"\"!\"#5" }{TEXT 20 0 "" }}}{EXCHG {PARA 296 "> " 0 "" {MPLTEXT 1 304 13 "X[1]:=X[0]+h;" }{MPLTEXT 1 304 0 "" } }{PARA 297 "" 1 "" {XPPMATH 20 "6#>&I\"XG6\"6#\"\"\"$\"+++++?!#5" } {TEXT 20 0 "" }}}{EXCHG {PARA 295 "" 0 "" {TEXT 302 9 "Here the " } {TEXT 303 1 "k" }{TEXT 302 28 " values are calculated. The " }{TEXT 303 2 "ky" }{TEXT 302 25 " values use function #1, " }{TEXT 303 2 "f1" }{TEXT 302 23 " for estimates and the " }{TEXT 303 2 "kz" }{TEXT 302 25 " values use function #2, " }{TEXT 303 2 "f2" }{TEXT 302 16 ". Note how each " }{TEXT 303 1 "k" }{TEXT 302 42 " values is based in part o n the preceding " }{TEXT 303 1 "k" }{TEXT 302 7 " value." }{TEXT 302 0 "" }}}{EXCHG {PARA 296 "> " 0 "" {MPLTEXT 1 304 24 "k1y:=f1(X[0],Y[0 ],Z[0]);" }{MPLTEXT 1 304 0 "" }}{PARA 297 "" 1 "" {XPPMATH 20 "6#>I$k 1yG6\"\"#5" }{TEXT 20 0 "" }}}{EXCHG {PARA 296 "> " 0 "" {MPLTEXT 1 304 24 "k1z:=f2(X[0],Y[0],Z[0]);" }{MPLTEXT 1 304 0 "" }}{PARA 297 "" 1 "" {XPPMATH 20 "6#>I$k1zG6\"#!$2\"\"\"#" }{TEXT 20 0 "" }}}{EXCHG {PARA 296 "> " 0 "" {MPLTEXT 1 304 50 "k2y:=f1(X[0]+0.5*h,Y[0]+0.5*k1y *h,Z[0]+0.5*k1z*h);" }{MPLTEXT 1 304 0 "" }}{PARA 297 "" 1 "" {XPPMATH 20 "6#>I$k2yG6\"$\"++++]Y!\"*" }{TEXT 20 0 "" }}}{EXCHG {PARA 296 "> " 0 "" {MPLTEXT 1 304 50 "k2z:=f2(X[0]+0.5*h,Y[0]+0.5*k1y *h,Z[0]+0.5*k1z*h);" }{MPLTEXT 1 304 0 "" }}{PARA 297 "" 1 "" {XPPMATH 20 "6#>I$k2zG6\"$!++++!e%!\")" }{TEXT 20 0 "" }}}{EXCHG {PARA 296 "> " 0 "" {MPLTEXT 1 304 50 "k3y:=f1(X[0]+0.5*h,Y[0]+0.5*k2y *h,Z[0]+0.5*k2z*h);" }{MPLTEXT 1 304 0 "" }}{PARA 297 "" 1 "" {XPPMATH 20 "6#>I$k3yG6\"$\"++++?a!\"*" }{TEXT 20 0 "" }}}{EXCHG {PARA 296 "> " 0 "" {MPLTEXT 1 304 50 "k3z:=f2(X[0]+0.5*h,Y[0]+0.5*k2y *h,Z[0]+0.5*k2z*h);" }{MPLTEXT 1 304 0 "" }}{PARA 297 "" 1 "" {XPPMATH 20 "6#>I$k3zG6\"$!+++]tX!\")" }{TEXT 20 0 "" }}}{EXCHG {PARA 296 "> " 0 "" {MPLTEXT 1 304 38 "k4y:=f1(X[0]+h,Y[0]+k3y*h,Z[0]+k3z*h) ;" }{MPLTEXT 1 304 0 "" }}{PARA 297 "" 1 "" {XPPMATH 20 "6#>I$k4yG6\"$ \"*+++`)!\"*" }{TEXT 20 0 "" }}}{EXCHG {PARA 296 "> " 0 "" {MPLTEXT 1 304 38 "k4z:=f2(X[0]+h,Y[0]+k3y*h,Z[0]+k3z*h);" }{MPLTEXT 1 304 0 "" } }{PARA 297 "" 1 "" {XPPMATH 20 "6#>I$k4zG6\"$!+++!e%Q!\")" }{TEXT 20 0 "" }}}{EXCHG {PARA 304 "" 0 "" {TEXT 309 131 "With all the k values \+ calculated we can solve for the solution at the next step and the deri vative (used in next step calculation.)" }{TEXT 309 0 "" }}}{EXCHG {PARA 296 "> " 0 "" {MPLTEXT 1 304 39 "Y[1]:=Y[0]+1/6*(k1y+2*k2y+2*k3y +k4y)*h;" }{MPLTEXT 1 304 0 "" }}{PARA 297 "" 1 "" {XPPMATH 20 "6#>&I \"YG6\"6#\"\"\"$\"+++J.6!\")" }{TEXT 20 0 "" }}}{EXCHG {PARA 296 "> " 0 "" {MPLTEXT 1 304 39 "Z[1]:=Z[0]+1/6*(k1z+2*k2z+2*k3z+k4z)*h;" } {MPLTEXT 1 304 0 "" }}{PARA 297 "" 1 "" {XPPMATH 20 "6#>&I\"ZG6\"6#\" \"\"$\"*++SK)!\"*" }{TEXT 20 0 "" }}}}{SECT 0 {PARA 302 "" 0 "" {TEXT 308 6 "Step 2" }{TEXT 308 0 "" }}{EXCHG {PARA 296 "> " 0 "" {MPLTEXT 1 304 13 "X[2]:=X[1]+h;" }{MPLTEXT 1 304 0 "" }}{PARA 297 "" 1 "" {XPPMATH 20 "6#>&I\"XG6\"6#\"\"#$\"+++++S!#5" }{TEXT 20 0 "" }}} {EXCHG {PARA 296 "> " 0 "" {MPLTEXT 1 304 24 "k1y:=f1(X[1],Y[1],Z[1]); " }{MPLTEXT 1 304 0 "" }}{PARA 297 "" 1 "" {XPPMATH 20 "6#>I$k1yG6\"$ \"*++SK)!\"*" }{TEXT 20 0 "" }}}{EXCHG {PARA 296 "> " 0 "" {MPLTEXT 1 304 24 "k1z:=f2(X[1],Y[1],Z[1]);" }{MPLTEXT 1 304 0 "" }}{PARA 297 "" 1 "" {XPPMATH 20 "6#>I$k1zG6\"$!+++TEQ!\")" }{TEXT 20 0 "" }}}{EXCHG {PARA 296 "> " 0 "" {MPLTEXT 1 304 50 "k2y:=f1(X[1]+0.5*h,Y[1]+0.5*k1y *h,Z[1]+0.5*k1z*h);" }{MPLTEXT 1 304 0 "" }}{PARA 297 "" 1 "" {XPPMATH 20 "6#>I$k2yG6\"$!+++,%*H!\"*" }{TEXT 20 0 "" }}}{EXCHG {PARA 296 "> " 0 "" {MPLTEXT 1 304 50 "k2z:=f2(X[1]+0.5*h,Y[1]+0.5*k1y *h,Z[1]+0.5*k1z*h);" }{MPLTEXT 1 304 0 "" }}{PARA 297 "" 1 "" {XPPMATH 20 "6#>I$k2zG6\"$!+++5'3$!\")" }{TEXT 20 0 "" }}}{EXCHG {PARA 296 "> " 0 "" {MPLTEXT 1 304 50 "k3y:=f1(X[1]+0.5*h,Y[1]+0.5*k2y *h,Z[1]+0.5*k2z*h);" }{MPLTEXT 1 304 0 "" }}{PARA 297 "" 1 "" {XPPMATH 20 "6#>I$k3yG6\"$!+++q`A!\"*" }{TEXT 20 0 "" }}}{EXCHG {PARA 296 "> " 0 "" {MPLTEXT 1 304 50 "k3z:=f2(X[1]+0.5*h,Y[1]+0.5*k2y*h,Z[1 ]+0.5*k2z*h);" }{MPLTEXT 1 304 0 "" }}{PARA 297 "" 1 "" {XPPMATH 20 "6 #>I$k3zG6\"$!++(p$>J!\")" }{TEXT 20 0 "" }}}{EXCHG {PARA 296 "> " 0 "" {MPLTEXT 1 304 38 "k4y:=f1(X[1]+h,Y[1]+k3y*h,Z[1]+k3z*h);" }{MPLTEXT 1 304 0 "" }}{PARA 297 "" 1 "" {XPPMATH 20 "6#>I$k4yG6\"$!++%RjS&!\"*" }{TEXT 20 0 "" }}}{EXCHG {PARA 296 "> " 0 "" {MPLTEXT 1 304 38 "k4z:= f2(X[1]+h,Y[1]+k3y*h,Z[1]+k3z*h);" }{MPLTEXT 1 304 0 "" }}{PARA 297 "" 1 "" {XPPMATH 20 "6#>I$k4zG6\"$!+?,WVC!\")" }{TEXT 20 0 "" }}}{EXCHG {PARA 296 "> " 0 "" {MPLTEXT 1 304 39 "Y[2]:=Y[1]+1/6*(k1y+2*k2y+2*k3y +k4y)*h;" }{MPLTEXT 1 304 0 "" }}{PARA 297 "" 1 "" {XPPMATH 20 "6#>&I \"YG6\"6#\"\"#$\"+-)yI0\"!\")" }{TEXT 20 0 "" }}}{EXCHG {PARA 296 "> " 0 "" {MPLTEXT 1 304 39 "Z[2]:=Z[1]+1/6*(k1z+2*k2z+2*k3z+k4z)*h;" } {MPLTEXT 1 304 0 "" }}{PARA 297 "" 1 "" {XPPMATH 20 "6#>&I\"ZG6\"6#\" \"#$!+S)HXR&!\"*" }{TEXT 20 0 "" }}}}{SECT 0 {PARA 302 "" 0 "" {TEXT 308 6 "Step 3" }{TEXT 308 0 "" }}{EXCHG {PARA 296 "> " 0 "" {MPLTEXT 1 304 13 "X[3]:=X[2]+h;" }{MPLTEXT 1 304 0 "" }}{PARA 297 "" 1 "" {XPPMATH 20 "6#>&I\"XG6\"6#\"\"$$\"+++++g!#5" }{TEXT 20 0 "" }}} {EXCHG {PARA 296 "> " 0 "" {MPLTEXT 1 304 24 "k1y:=f1(X[2],Y[2],Z[2]); " }{MPLTEXT 1 304 0 "" }}{PARA 297 "" 1 "" {XPPMATH 20 "6#>I$k1yG6\"$! +S)HXR&!\"*" }{TEXT 20 0 "" }}}{EXCHG {PARA 296 "> " 0 "" {MPLTEXT 1 304 24 "k1z:=f2(X[2],Y[2],Z[2]);" }{MPLTEXT 1 304 0 "" }}{PARA 297 "" 1 "" {XPPMATH 20 "6#>I$k1zG6\"$!+Q/LIC!\")" }{TEXT 20 0 "" }}}{EXCHG {PARA 296 "> " 0 "" {MPLTEXT 1 304 50 "k2y:=f1(X[2]+0.5*h,Y[2]+0.5*k1y *h,Z[2]+0.5*k1z*h);" }{MPLTEXT 1 304 0 "" }}{PARA 297 "" 1 "" {XPPMATH 20 "6#>I$k2yG6\"$!+y-'[#y!\"*" }{TEXT 20 0 "" }}}{EXCHG {PARA 296 "> " 0 "" {MPLTEXT 1 304 50 "k2z:=f2(X[2]+0.5*h,Y[2]+0.5*k1y *h,Z[2]+0.5*k1z*h);" }{MPLTEXT 1 304 0 "" }}{PARA 297 "" 1 "" {XPPMATH 20 "6#>I$k2zG6\"$!+c%GCy\"!\")" }{TEXT 20 0 "" }}}{EXCHG {PARA 296 "> " 0 "" {MPLTEXT 1 304 50 "k3y:=f1(X[2]+0.5*h,Y[2]+0.5*k2y *h,Z[2]+0.5*k2z*h);" }{MPLTEXT 1 304 0 "" }}{PARA 297 "" 1 "" {XPPMATH 20 "6#>I$k3yG6\"$!+'Hep<(!\"*" }{TEXT 20 0 "" }}}{EXCHG {PARA 296 "> " 0 "" {MPLTEXT 1 304 50 "k3z:=f2(X[2]+0.5*h,Y[2]+0.5*k2y *h,Z[2]+0.5*k2z*h);" }{MPLTEXT 1 304 0 "" }}{PARA 297 "" 1 "" {XPPMATH 20 "6#>I$k3zG6\"$!+Q*)4R=!\")" }{TEXT 20 0 "" }}}{EXCHG {PARA 296 "> " 0 "" {MPLTEXT 1 304 38 "k4y:=f1(X[2]+h,Y[2]+k3y*h,Z[2]+ k3z*h);" }{MPLTEXT 1 304 0 "" }}{PARA 297 "" 1 "" {XPPMATH 20 "6#>I$k4 yG6\"$!+;xss!*!\"*" }{TEXT 20 0 "" }}}{EXCHG {PARA 296 "> " 0 "" {MPLTEXT 1 304 38 "k4z:=f2(X[2]+h,Y[2]+k3y*h,Z[2]+k3z*h);" }{MPLTEXT 1 304 0 "" }}{PARA 297 "" 1 "" {XPPMATH 20 "6#>I$k4zG6\"$!+mL2k7!\")" }{TEXT 20 0 "" }}}{EXCHG {PARA 296 "> " 0 "" {MPLTEXT 1 304 39 "Y[3]:= Y[2]+1/6*(k1y+2*k2y+2*k3y+k4y)*h;" }{MPLTEXT 1 304 0 "" }}{PARA 297 "" 1 "" {XPPMATH 20 "6#>&I\"YG6\"6#\"\"$$\"+j[U[!*!\"*" }{TEXT 20 0 "" } }}{EXCHG {PARA 296 "> " 0 "" {MPLTEXT 1 304 39 "Z[3]:=Z[2]+1/6*(k1z+2* k2z+2*k3z+k4z)*h;" }{MPLTEXT 1 304 0 "" }}{PARA 297 "" 1 "" {XPPMATH 20 "6#>&I\"ZG6\"6#\"\"$$!+q$\\./*!\"*" }{TEXT 20 0 "" }}}}{SECT 0 {PARA 302 "" 0 "" {TEXT 308 6 "Step 4" }{TEXT 308 0 "" }}{EXCHG {PARA 296 "> " 0 "" {MPLTEXT 1 304 13 "X[4]:=X[3]+h;" }{MPLTEXT 1 304 0 "" } }{PARA 297 "" 1 "" {XPPMATH 20 "6#>&I\"XG6\"6#\"\"%$\"+++++!)!#5" } {TEXT 20 0 "" }}}{EXCHG {PARA 296 "> " 0 "" {MPLTEXT 1 304 24 "k1y:=f1 (X[3],Y[3],Z[3]);" }{MPLTEXT 1 304 0 "" }}{PARA 297 "" 1 "" {XPPMATH 20 "6#>I$k1yG6\"$!+q$\\./*!\"*" }{TEXT 20 0 "" }}}{EXCHG {PARA 296 "> \+ " 0 "" {MPLTEXT 1 304 24 "k1z:=f2(X[3],Y[3],Z[3]);" }{MPLTEXT 1 304 0 "" }}{PARA 297 "" 1 "" {XPPMATH 20 "6#>I$k1zG6\"$!+&edkD\"!\")" } {TEXT 20 0 "" }}}{EXCHG {PARA 296 "> " 0 "" {MPLTEXT 1 304 50 "k2y:=f1 (X[3]+0.5*h,Y[3]+0.5*k1y*h,Z[3]+0.5*k1z*h);" }{MPLTEXT 1 304 0 "" }} {PARA 297 "" 1 "" {XPPMATH 20 "6#>I$k2yG6\"$!+'p!oH5!\")" }{TEXT 20 0 "" }}}{EXCHG {PARA 296 "> " 0 "" {MPLTEXT 1 304 50 "k2z:=f2(X[3]+0.5*h ,Y[3]+0.5*k1y*h,Z[3]+0.5*k1z*h);" }{MPLTEXT 1 304 0 "" }}{PARA 297 "" 1 "" {XPPMATH 20 "6#>I$k2zG6\"$!+gebRt!\"*" }{TEXT 20 0 "" }}}{EXCHG {PARA 296 "> " 0 "" {MPLTEXT 1 304 50 "k3y:=f1(X[3]+0.5*h,Y[3]+0.5*k2y *h,Z[3]+0.5*k2z*h);" }{MPLTEXT 1 304 0 "" }}{PARA 297 "" 1 "" {XPPMATH 20 "6#>I$k3yG6\"$!+c\\Iu(*!\"*" }{TEXT 20 0 "" }}}{EXCHG {PARA 296 "> " 0 "" {MPLTEXT 1 304 50 "k3z:=f2(X[3]+0.5*h,Y[3]+0.5*k2y *h,Z[3]+0.5*k2z*h);" }{MPLTEXT 1 304 0 "" }}{PARA 297 "" 1 "" {XPPMATH 20 "6#>I$k3zG6\"$!+!fAw+)!\"*" }{TEXT 20 0 "" }}}{EXCHG {PARA 296 "> " 0 "" {MPLTEXT 1 304 38 "k4y:=f1(X[3]+h,Y[3]+k3y*h,Z[3]+ k3z*h);" }{MPLTEXT 1 304 0 "" }}{PARA 297 "" 1 "" {XPPMATH 20 "6#>I$k4 yG6\"$!+*Q(=k5!\")" }{TEXT 20 0 "" }}}{EXCHG {PARA 296 "> " 0 "" {MPLTEXT 1 304 38 "k4z:=f2(X[3]+h,Y[3]+k3y*h,Z[3]+k3z*h);" }{MPLTEXT 1 304 0 "" }}{PARA 297 "" 1 "" {XPPMATH 20 "6#>I$k4zG6\"$!+NQ%p\\$!\"* " }{TEXT 20 0 "" }}}{EXCHG {PARA 296 "> " 0 "" {MPLTEXT 1 304 39 "Y[4] :=Y[3]+1/6*(k1y+2*k2y+2*k3y+k4y)*h;" }{MPLTEXT 1 304 0 "" }}{PARA 297 "" 1 "" {XPPMATH 20 "6#>&I\"YG6\"6#\"\"%$\"+EmFaq!\"*" }{TEXT 20 0 "" }}}{EXCHG {PARA 296 "> " 0 "" {MPLTEXT 1 304 39 "Z[4]:=Z[3]+1/6*(k1z+2 *k2z+2*k3z+k4z)*h;" }{MPLTEXT 1 304 0 "" }}{PARA 297 "" 1 "" {XPPMATH 20 "6#>&I\"ZG6\"6#\"\"%$!+fy))f5!\")" }{TEXT 20 0 "" }}}}{SECT 0 {PARA 302 "" 0 "" {TEXT 308 6 "Step 5" }{TEXT 308 0 "" }}{EXCHG {PARA 296 "> " 0 "" {MPLTEXT 1 304 13 "X[5]:=X[4]+h;" }{MPLTEXT 1 304 0 "" } }{PARA 297 "" 1 "" {XPPMATH 20 "6#>&I\"XG6\"6#\"\"&$\"+++++5!\"*" } {TEXT 20 0 "" }}}{EXCHG {PARA 296 "> " 0 "" {MPLTEXT 1 304 24 "k1y:=f1 (X[4],Y[4],Z[4]);" }{MPLTEXT 1 304 0 "" }}{PARA 297 "" 1 "" {XPPMATH 20 "6#>I$k1yG6\"$!+fy))f5!\")" }{TEXT 20 0 "" }}}{EXCHG {PARA 296 "> " 0 "" {MPLTEXT 1 304 24 "k1z:=f2(X[4],Y[4],Z[4]);" }{MPLTEXT 1 304 0 " " }}{PARA 297 "" 1 "" {XPPMATH 20 "6#>I$k1zG6\"$!++F2lM!\"*" }{TEXT 20 0 "" }}}{EXCHG {PARA 296 "> " 0 "" {MPLTEXT 1 304 50 "k2y:=f1(X[4]+ 0.5*h,Y[4]+0.5*k1y*h,Z[4]+0.5*k1z*h);" }{MPLTEXT 1 304 0 "" }}{PARA 297 "" 1 "" {XPPMATH 20 "6#>I$k2yG6\"$!+'eQX4\"!\")" }{TEXT 20 0 "" }} }{EXCHG {PARA 296 "> " 0 "" {MPLTEXT 1 304 50 "k2z:=f2(X[4]+0.5*h,Y[4] +0.5*k1y*h,Z[4]+0.5*k1z*h);" }{MPLTEXT 1 304 0 "" }}{PARA 297 "" 1 "" {XPPMATH 20 "6#>I$k2zG6\"$\"++U0wS!#5" }{TEXT 20 0 "" }}}{EXCHG {PARA 296 "> " 0 "" {MPLTEXT 1 304 50 "k3y:=f1(X[4]+0.5*h,Y[4]+0.5*k2y*h,Z[4 ]+0.5*k2z*h);" }{MPLTEXT 1 304 0 "" }}{PARA 297 "" 1 "" {XPPMATH 20 "6 #>I$k3yG6\"$!+0=\"e0\"!\")" }{TEXT 20 0 "" }}}{EXCHG {PARA 296 "> " 0 "" {MPLTEXT 1 304 50 "k3z:=f2(X[4]+0.5*h,Y[4]+0.5*k2y*h,Z[4]+0.5*k2z*h );" }{MPLTEXT 1 304 0 "" }}{PARA 297 "" 1 "" {XPPMATH 20 "6#>I$k3zG6\" $!++-yHE!#5" }{TEXT 20 0 "" }}}{EXCHG {PARA 296 "> " 0 "" {MPLTEXT 1 304 38 "k4y:=f1(X[4]+h,Y[4]+k3y*h,Z[4]+k3z*h);" }{MPLTEXT 1 304 0 "" } }{PARA 297 "" 1 "" {XPPMATH 20 "6#>I$k4yG6\"$!+>u9l5!\")" }{TEXT 20 0 "" }}}{EXCHG {PARA 296 "> " 0 "" {MPLTEXT 1 304 38 "k4z:=f2(X[4]+h,Y[4 ]+k3y*h,Z[4]+k3z*h);" }{MPLTEXT 1 304 0 "" }}{PARA 297 "" 1 "" {XPPMATH 20 "6#>I$k4zG6\"$\"+I$*)\\(H!\"*" }{TEXT 20 0 "" }}}{EXCHG {PARA 296 "> " 0 "" {MPLTEXT 1 304 39 "Y[5]:=Y[4]+1/6*(k1y+2*k2y+2*k3y +k4y)*h;" }{MPLTEXT 1 304 0 "" }}{PARA 297 "" 1 "" {XPPMATH 20 "6#>&I \"YG6\"6#\"\"&$\"+1YO7\\!\"*" }{TEXT 20 0 "" }}}{EXCHG {PARA 296 "> " 0 "" {MPLTEXT 1 304 39 "Z[5]:=Z[4]+1/6*(k1z+2*k2z+2*k3z+k4z)*h;" } {MPLTEXT 1 304 0 "" }}{PARA 297 "" 1 "" {XPPMATH 20 "6#>&I\"ZG6\"6#\" \"&$!+)Gd01\"!\")" }{TEXT 20 0 "" }}}}{SECT 0 {PARA 302 "" 0 "" {TEXT 308 7 "Results" }{TEXT 308 0 "" }}{EXCHG {PARA 295 "" 0 "" {TEXT 302 24 "Approximate solution at " }{TEXT 303 6 "x = xf" }{TEXT 302 0 "" }} }{EXCHG {PARA 296 "> " 0 "" {MPLTEXT 1 304 5 "Y[5];" }{MPLTEXT 1 304 0 "" }}{PARA 297 "" 1 "" {XPPMATH 20 "6#$\"+1YO7\\!\"*" }{TEXT 20 0 "" }}}{EXCHG {PARA 295 "" 0 "" {TEXT 302 18 "Exact solution at " }{TEXT 303 6 "x = xf" }{TEXT 302 0 "" }}}{EXCHG {PARA 305 "> " 0 "" {MPLTEXT 1 310 4 "yf:=" }{MPLTEXT 1 310 23 "evalf(subs(x=xf,y(x)));" }{MPLTEXT 1 310 0 "" }}{PARA 297 "" 1 "" {XPPMATH 20 "6#>I#yfG6\"$\"+pAn7\\!\"*" }{TEXT 20 0 "" }}}{EXCHG {PARA 304 "" 0 "" {TEXT 309 39 "Absolute Rel ative True Error Percentage" }{TEXT 309 0 "" }}}{EXCHG {PARA 305 "> " 0 "" {MPLTEXT 1 310 22 "abs((Y[5]-yf)/yf)*100;" }{MPLTEXT 1 310 0 "" } }{PARA 297 "" 1 "" {XPPMATH 20 "6#$\"+O/kii!#7" }{TEXT 20 0 "" }}} {EXCHG {PARA 304 "" 0 "" {TEXT 309 35 "Graphical depiction of the solu tion" }{TEXT 309 0 "" }}}{EXCHG {PARA 296 "> " 0 "" {MPLTEXT 1 304 32 "data:=[seq([X[i],Y[i]],i=0..5)]:" }{MPLTEXT 1 304 0 "" }}}{EXCHG {PARA 296 "> " 0 "" {MPLTEXT 1 304 147 "plot([y(x),data],x=x0..xf,colo 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lFa[l-Ff[l6#Q.ApproximationFi[l-%&TITLEG6#Q?Exact~and~Approximate~Solu tionFi[l-%+AXESLABELSG6$Q\"xFi[lQ!Fi[l-%*THICKNESSG6#\"\"#-%%VIEWG6$;F `[lF^[l;$\"1#yXN:#z*y%!#:$\"1;l6ofN;6!#9" 1 2 2 0 10 2 2 6 1 4 2 1.0 45.0 45.0 1 1 "Exact" "Approximation" }}{TEXT 307 0 "" }}}}{PARA 306 " " 0 "" {TEXT 294 0 "" }}}{SECT 0 {PARA 293 "" 0 "" {TEXT -1 10 "Refere nces" }{TEXT -1 0 "" }}{EXCHG {PARA 307 "" 0 "" {TEXT 241 17 "Chapra & Canale, " }{TEXT 303 64 "Numerical Methods for Engineering: With Prog ramming Applications" }{TEXT 241 32 ". 3rd ed. WCB/McGraw-Hill. 1998." }{TEXT 270 0 "" }}}}{SECT 0 {PARA 293 "" 0 "" {TEXT -1 10 "Conclusion " }{TEXT -1 0 "" }}{EXCHG {PARA 308 "" 0 "" {TEXT 241 309 "This sheet \+ has showed us how we can use the methods learned for solving first ord er ordinary differential equaitons and apply them to solve higher orde r problems. In this example we just did a second order ordinary differ ential equation, but the same method could be utilitized to solve high er orders as well." }{TEXT 241 0 "" }}}}{PARA 309 "" 0 "" {TEXT 279 0 "" }}{PARA 309 "" 0 "" {TEXT 279 398 "Legal Notice: The copyright for \+ this application is owned by the author(s). Neither Maplesoft nor the \+ author are responsible for any errors contained within and are not lia ble for any damages resulting from the use of this material. This appl ication is intended for non-commercial, non-profit use only. Contact t he author for permission if you wish to use this application in for-pr ofit activities." }{TEXT 279 0 "" }}{PARA 310 "" 0 "" {TEXT 258 0 "" } }{PARA 311 "" 0 "" {TEXT 311 0 "" }}{PARA 312 "" 0 "" {TEXT -1 0 "" }} }{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }