{VERSION 4 0 "SUN SPARC SOLARIS" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Warning" 2 7 1 {CSTYLE "" -1 -1 " " 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE " " 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 13 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 256 "" 0 "" {TEXT 256 11 "Math 2280-2" }}{PARA 257 "" 0 "" {TEXT 257 24 "Maple Project 1, Part 1." }}{PARA 258 "" 0 "" {TEXT -1 16 "January 23, 2001" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 " This project is available on-line at our course Maple page, " }{TEXT 259 64 "http://www.math.utah.edu/~korevaar/2280s pring01/2280maple.html " }{TEXT -1 432 " You can also download the an swer template from that location. If you are shaky about Maple or th e computer-lab, there is a tutorial (really written for Math 2250 stud ents) which you can also download and work on, available from the same location. Always make sure that you download the \".mws\" files to \+ open from the Maple window. If you want versions to print out or to l ook at from your browser, try the \".pdf\" files as well." }}{PARA 0 " " 0 "" {TEXT -1 144 " This project has two parts. In Part 1 you w ill study logistic equation applications from the computer project of \+ section 2.1 in the text, " }{TEXT 260 50 "Differential Equations and B oundary Value Problems" }{TEXT -1 343 ", second edition, by Edwards-Pe nney. In Part 2 of your Maple project you will study numerical metho ds for the solution of first order differential equations, using ideas from sections 2.4-2.6. Each part of the project has a solution temp late into which you will do your work. These templates are also avail able at the course Maple page. " }}{PARA 259 "" 0 "" {TEXT -1 309 " \+ It is very important when you read Math in the text, here, or elsew here, that you read slowly, sentence by sentence, making sure to think about and grasp each concept before moving on to the next one. If yo u try to read mathematics as quickly as you read the newspaper you wil l become lost very quickly." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 258 9 "Part 1: " }{TEXT -1 85 "Recall the logistic eq uation for population change which you have just been studying:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%#dPG\"\"\"%#dtG!\"\"*&%#kPGF&,&%\" MGF&%\"PGF(F&" }}{PARA 0 "" 0 "" {TEXT -1 61 "or, following the text a nd its equation numbering at page 86:" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#>%&eqtn1G/*&%#dPG\"\"\"%#dtG!\"\",&*&%\"aGF(%\"PGF(F(*& %\"bGF()F.\"\"#F(F(" }}}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 954 "with a=kM and b=-k. In our model the parameters k,M, a,b were related to assumptions about birth and death rates. Suppose \+ you have a real population and want to pick parameters a and b to make a good model. One way would be to try to estimate fertility and morb idity rates based on birth and death data, but that could get quite co mplicated. For example, if you want to develop an accurate model of w orld population growth based on this sort of analysis you would probab ly need to collect data from different regions of our planet and devel op different parameters for different societies, solve the problem in \+ each part of the globe and then add your results together. A more si mple-minded approach is to just assume the logistic model and to use e xisting population data to deduce good choices for a and b. The book \+ explains a good way to do this on pages 86-87, which also checks along the way whether the model seems to be justified by the data:" }} {PARA 0 "" 0 "" {TEXT -1 62 " If you divide the logistic DE, eqtn1 above, by P, you get" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#>%&eqt n2G/*&7#*&\"\"\"F)%\"PG!\"\"F)7#*&%#dPGF)%#dtGF+F),&%\"aGF)*&%\"bGF)F* F)F)" }}}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 565 "If you have multi-year population data you can get good estimates for the left side of eqtn2 by using difference quotients to estimate \+ dP/dt. Dividing by P gives an estimate for (dP/dt)/P. If you can do \+ this for a lot of different times you can get a collection of points [ P, (1/P)(dP/dt)]. If these points seem to lie approximately along a l ine, then eqtn2 will be a good model for the population problem, and y ou can estimate the parameters \"a\" and \"b\" by getting the vertical axis-intercept and slope of the line which best fits the point data, \+ respectively. " }}{PARA 0 "" 0 "" {TEXT -1 557 " For example, look ing at the USA data in Figure 2.1.7 on page 87, we can estimate dP/dt \+ in 1800 in a \"centered\" way by taking the difference (P(1810)-P(179 0))/20 (in units of people/year). Centered differences as in (eqtn3 ) page 86, generally give more accurate estimates for the derivative \+ than the one-sided differences used in the limit definition. This is \+ shown geometrically in Figure 2.1.6, page 87. Finally we would divide our centered estimate for dP/dt in 1800 by P(1800) to get an estimate for (dP/dt)/P in 1800, for the USA population:" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 8 "restart:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 314 "\nP1:=5.308;\n #population in 1800, in millions.\n #Remember you ca n do multiline commands\n #by holding down the shift key when you hit return\n #or enter.\nP1prime:=(7.240-3.929)/20;\n #estimate for dP/ dt in 1800, in millions of\n #people per year, see page 86 bottom\n \+ #and top right entry in Figure 2.1.7 page 87" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#P1G$\"%3`!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%( P1primeG$\"+++]b;!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "P1p rimeoverP1:=P1prime/P1;\n #estimate for (1/P)*(dP/dt) in 1800;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%.P1primeoverP1G$\"+nr()=J!#6" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "point1:=[P1,P1primeoverP1]; \n #the left-most point on the graph of Figure 2.1.8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'point1G7$$\"%3`!\"$$\"+nr()=J!#6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 164 "W e can automate this process. We are still using the table on page 87, and the one on pate 80. You should verify how these numbers were ext racted from the tables." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "w ith(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected na mes norm and trace have been redefined and unprotected\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 380 "pops:=matrix(21,2,[[1790,3.9],[180 0,5.3],[1810,7.2],\n [1820,9.6],[1830,12.9],[1840,17.1],\n \+ [1850,23.2],[1860,31.4],[1870,38.6],\n [1880,50.2],[1890,63.0], [1900,76.2],\n [1910,92.2],[1920,106.0],[1930,123.2],\n [1 940,132.2],[1950,151.3],[1960,179.3],\n [1970,203.3],[1980,225.6 ],[1990,248.7]]);\n #matrix of populations \n \+ " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%popsG-%'matrixG6#777$\"%!z\"$\"#R!\"\"7$\"%+=$\"#`F- 7$\"%5=$\"#sF-7$\"%?=$\"#'*F-7$\"%I=$\"$H\"F-7$\"%S=$\"$r\"F-7$\"%]=$ \"$K#F-7$\"%g=$\"$9$F-7$\"%q=$\"$'QF-7$\"%!)=$\"$-&F-7$\"%!*=$\"$I'F-7 $\"%+>$\"$i(F-7$\"%5>$\"$A*F-7$\"%?>$\"%g5F-7$\"%I>$\"%K7F-7$\"%S>$\"% A8F-7$\"%]>$\"%8:F-7$\"%g>$\"%$z\"F-7$\"%q>$\"%L?F-7$\"%!)>$\"%cAF-7$ \"%!*>$\"%([#F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "for i fr om 1 to 11 do\nlspoints[i]:=[pops[i+1,2],\n (pops[i+2,2]-pops[i,2])/ (20*pops[i+1,2])];\nod;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%)lspointsG6#\"\"\"7$$\"#`!\"\"$\"+Z v?8J!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%)lspointsG6#\"\"#7$$\"#s !\"\"$\"+666')H!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%)lspointsG6# \"\"$7$$\"#'*!\"\"$\"+++voH!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%) lspointsG6#\"\"%7$$\"$H\"!\"\"$\"+Wn(p!H!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%)lspointsG6#\"\"&7$$\"$r\"!\"\"$\"+1fp6I!#6" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%)lspointsG6#\"\"'7$$\"$K#!\"\"$\"+_ l*=3$!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%)lspointsG6#\"\"(7$$\"$ 9$!\"\"$\"++$HAX#!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%)lspointsG6 #\"\")7$$\"$'Q!\"\"$\"+gJBNC!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&% )lspointsG6#\"\"*7$$\"$-&!\"\"$\"+%))y-V#!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%)lspointsG6#\"#57$$\"$I'!\"\"$\"+k?\\j?!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%)lspointsG6#\"#67$$\"$i(!\"\"$\"+)\\5g\"> !#6" }}}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 201 "We will follow investigation A and find the least squares line fit t o this data, in order to extract values for \"a\" and \"b\" in eqtn2. \+ We remember how to do this from Math 2270, section 8.4 of Kolman." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "A:=matrix(11,2);\nB:=vector( 11);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%&arrayG6%;\"\"\"\"#6;F )\"\"#7\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG-%&arrayG6$;\"\"\" \"#67\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "for i from 1 to \+ 11 do\nA[i,1]:=lspoints[i][1]:\nA[i,2]:=1:\nB[i]:=lspoints[i][2]:\nod: \n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "evalm(A);evalm(B);\n \+ #check work" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7-7$$\"#`! \"\"\"\"\"7$$\"#sF*F+7$$\"#'*F*F+7$$\"$H\"F*F+7$$\"$r\"F*F+7$$\"$K#F*F +7$$\"$9$F*F+7$$\"$'QF*F+7$$\"$-&F*F+7$$\"$I'F*F+7$$\"$i(F*F+" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7-$\"+Zv?8J!#6$\"+666')HF) $\"+++voHF)$\"+Wn(p!HF)$\"+1fp6IF)$\"+_l*=3$F)$\"++$HAX#F)$\"+gJBNCF)$ \"+%))y-V#F)$\"+k?\\j?F)$\"+)\\5g\">F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "linsolve(transpose(A)&*(A),transpose(A)&*B);\n #leas t squares solution\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7$ $!++o8%p\"!#8$\"+R_5&=$!#6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "a:=.3185105239e-1;\n #intercept\nb:=-.1694136800e-3;\n #slope " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG$\"+R_5&=$!#6" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"bG$!++o8%p\"!#8" }}}{PARA 0 "" 0 "" {TEXT -1 31 "Now we can create figure 2.1.8:" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "with(plots): #load the p lotting package" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name chan gecoords has been redefined\n" }}}{PARA 11 "" 1 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 240 "pict1:=pointplot(\{seq(\n \+ lspoints[i],i=1..11)\}): \n #a plot of the points,\n #with outp ut suppressed. Make sure\n #to end this command with a colon! If y ou use a\n #semicolon you get a huge mess when you have\n #a lot of points " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "line:=plot(a +b*P,P=0..100, color=black): #same warning here\n #about using a co lon vs semicolon" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 156 "displa y(\{pict1,line\}, title=\"Figure 2.1.8\"); \n #now use a semicolon , and\n #get a picture containing the line and the \n #three poi nts we computed. " }}{PARA 13 "" 1 "" {GLPLOT2D 279 176 176 {PLOTDATA 2 "6'-%'POINTSG6-7$$\"#`!\"\"$\"+Zv?8J!#67$$\"#sF)$\"+666')HF,7$$\"#'* F)$\"+++voHF,7$$\"$H\"F)$\"+Wn(p!HF,7$$\"$r\"F)$\"+1fp6IF,7$$\"$K#F)$ \"+_l*=3$F,7$$\"$9$F)$\"++$HAX#F,7$$\"$'QF)$\"+gJBNCF,7$$\"$-&F)$\"+%) )y-V#F,7$$\"$I'F)$\"+k?\\j?F,7$$\"$i(F)$\"+)\\5g\">F,-%'CURVESG6$7S7$$ \"\"!F_o$\"3B+++R_5&=$!#>7$$\"3ymmm;arz@!#<$\"32x4*z(y<[JFbo7$$\"3eLL$ e9ui2%Ffo$\"39_liyv/;JFbo7$$\"3*omm;z_\"4iFfo$\"3.[\")e'p8*zIFbo7$$\"3 OommT&phN)Ffo$\"3yLUs&HSN/$Fbo7$$\"3KLLe*=)H\\5!#;$\"3-]`Gr(Rt+$Fbo7$$ \"3imm\"z/3uC\"F[q$\"3O@39^sxtHFbo7$$\"3.++DJ$RDX\"F[q$\"3d)>[b?D!RHFb o7$$\"3vmm\"zR'ok;F[q$\"3a$*HG!f%3.HFbo7$$\"39++D1J:w=F[q$\"3ocn-P#fs' GFbo7$$\"3oLLL3En$4#F[q$\"3m^qqd%3/$GFbo7$$\"3,nm;/RE&G#F[q$\"3)z&RAr- &zz#Fbo7$$\"3_+++D.&4]#F[q$\"3DaX%4/59w#Fbo7$$\"32+++vB_Zs#Fbo7$$\"3W+++v'Hi#HF[q$\"3,YIB,>O*o#Fbo7$$\"3%pm;z*ev:JF[q$\"3@LA NmNDdEFbo7$$\"3)RLL$347TLF[q$\"3[2Vf]O2>EFbo7$$\"3KLLLLY.KNF[q$\"3h\\& zQDIne#Fbo7$$\"3N++D\"o7Tv$F[q$\"3qqZ`%>2\"\\DFbo7$$\"3kLLL$Q*o]RF[q$ \"3[pD.7W!e^#Fbo7$$\"3m++D\"=lj;%F[q$\"3Xq0U\")fEzCFbo7$$\"3A++vV&RY2aF[q$\"3v,Z)e@2!pAFbo7$$ \"3%pmm\"zXu9cF[q$\"3#RK$e(p!*QB#Fbo7$$\"3F+++]y))GeF[q$\"3%>@C!)*=h(> #Fbo7$$\"3I++]i_QQgF[q$\"334@Uq,7i@Fbo7$$\"3>++D\"y%3TiF[q$\"3\"HW;!*4 !yF@Fbo7$$\"3l****\\P![hY'F[q$\"3EZBa/8l*3#Fbo7$$\"3MLLL$Qx$omF[q$\"3I H2'o)3Rb?Fbo7$$\"3$3++]P+V)oF[q$\"3)*peCy0\")=?Fbo7$$\"3@mm\"zpe*zqF[q $\"3[Hzt\"Qjc)>Fbo7$$\"3;,++D\\'QH(F[q$\"3?EGjSZU\\>Fbo7$$\"3yKLe9S8& \\(F[q$\"3/ki\\.qK:>Fbo7$$\"3!4+]i?=bq(F[q$\"3^)=sL/&oz=Fbo7$$\"33LLL3 s?6zF[q$\"3vBPf7&Q[%=Fbo7$$\"3g++DJXaE\")F[q$\"3\"G1xUUd$3=Fbo7$$\"3%z mmm'*RRL)F[q$\"3[AzM+=At(>yf\" Fbo7$$\"3J,+D\"oK0e*F[q$\"3Hq\"46%>.i:Fbo7$$\"3Y,+v=5s#y*F[q$\"3_8-!3Z yx_\"Fbo7$$\"$+\"F_o$\"31+++R%o4\\\"Fbo-%'COLOURG6&%$RGBGF_oF_oF_o-%&T ITLEG6#Q-Figure~2.1.86\"-%+AXESLABELSG6$Q\"P6\"Q!6\"-%%VIEWG6$;F^oFa^l %(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "C urve 1" "Curve 2" }}}}{PARA 260 "" 1 "" {TEXT -1 303 "Figure 2.1.8 sho ws that the logistic model is a pretty good one for the U.S. populatio n in the 1800's. Let's use the \"a\" and \"b\" which we found with th e least squares fit, use the population in 1900 for our initial condit ion, and see how the logistic model works when we try to extend it to \+ the 1900's:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "with(DEtools) : #load the DE package" }}{PARA 7 "" 1 "" {TEXT -1 45 "Warning, the n ame adjoint has been redefined\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "deqtn1:=diff(x(t),t)=a*x(t) + b*x(t)^2;\n #logist ic eqtn with our parameters" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'deqt n1G/-%%diffG6$-%\"xG6#%\"tGF,,&F)$\"+R_5&=$!#6*&$\"++o8%p\"!#8\"\"\")F )\"\"#F5!\"\"" }}}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 103 "P:=dsolve(\{deqtn1,x(0)=76.21\},x(t));\n \+ #take 1900 as t=0 and solve the initial value\n #problem " }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PG/-%\"xG6#%\"tG,$*&\"\"\"F,,&\")o 8%p\"F,*&#\"-s$e.S*=\"%@wF,-%$expG6#,$F)#!+R_5&=$\"-+++++5F,F,!\"\"\"+ R_5&=$" }}}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "f:=s->evalf(subs(t=s,rhs(P))); #extract the right-ha nd side\n #from the above expression to make your solution functi on" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"sG6\"6$%)operatorG%& arrowGF(-%&evalfG6#-%%subsG6$/%\"tG9$-%$rhsG6#%\"PGF(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "f(s); #check that those weird su bs and rhs commands really work\n " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"\"F%,&$\")o8%p\"\"\"!F%*&$\"+gGC&[#!\"#F%-%$expG6#,$%\"sG$! +R_5&=$!#6F%F%!\"\"$\"+R_5&=$F)" }}}{PARA 261 "" 1 "" {TEXT -1 105 " \+ We can see how the model works by plotting actual populations agains t predicted ones, for the 1900's." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 130 "actual:=pointplot(\{seq([po ps[i,1],pops[i,2]],i=1..21)\}):\nmodel:=plot(f(s-1900),s=1790..2000,co lor=black):\ndisplay(\{actual,model\});" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'POINTSG677$$\"%!z\"\"\"!$\"#R!\"\"7$$\"% +=F)$\"#`F,7$$\"%5=F)$\"#sF,7$$\"%?=F)$\"#'*F,7$$\"%I=F)$\"$H\"F,7$$\" %S=F)$\"$r\"F,7$$\"%]=F)$\"$K#F,7$$\"%g=F)$\"$9$F,7$$\"%q=F)$\"$'QF,7$ $\"%!)=F)$\"$-&F,7$$\"%!*=F)$\"$I'F,7$$\"%+>F)$\"$i(F,7$$\"%5>F)$\"$A* F,7$$\"%?>F)$\"%g5F,7$$\"%I>F)$\"%K7F,7$$\"%S>F)$\"%A8F,7$$\"%]>F)$\"% 8:F,7$$\"%g>F)$\"%$z\"F,7$$\"%q>F)$\"%L?F,7$$\"%!)>F)$\"%cAF,7$$\"%!*> F)$\"%([#F,-%'CURVESG6$7S7$F'$\"3\"Q-!4rzkyP!#<7$$\"3-+]P-ud%z\"!#9$\" 3xvdMiK*zN%Fbr7$$\"3+]iqv,c)z\"Ffr$\"3wa-bLC'>$\\Fbr7$$\"34+D'3ARI!=Ff r$\"3&G\"RJL8WlcFbr7$$\"3%**\\Pg&za2=Ffr$\"3?s;mqz15lFbr7$$\"3'*\\7)>E N?\"=Ffr$\"3\\4L#*yYbquFbr7$$\"3-]i+pb>;=Ffr$\"3(Qw+)Q7J\"[)Fbr7$$\"3 \"*\\i&fK.0#=Ffr$\"35,v81')\\k'*Fbr7$$\"3$*\\iN9%e\\#=Ffr$\"3KW3CdT806 !#;7$$\"33]7B:#*RH=Ffr$\"35eep3^xh7F\\u7$$\"3++]xCr'R$=Ffr$\"3gc\"4:a, UW\"F\\u7$$\"3\"**\\()>a!*z$=Ffr$\"3(RU\"[1bfC;F\\u7$$\"3,+]#o&*>D%=Ff r$\"3B'yo!fa)=&=F\\u7$$\"3-+]()pz1Z=Ffr$\"3;w-jW@=3@F\\u7$$\"35+]4% )[0UF\\u7$$\"32]i!)oO\\x=Ffr$\"3=`1l>>8&p%F\\u7$$\"32](=/`1=)=Ffr$\"3= @sFsSW&>&F\\u7$$\"3$**\\i!\\lI')=Ffr$\"36/*3,(3>^dF\\u7$$\"3'*\\7$4(*Q /*=Ffr$\"3qpigz*)e*G'F\\u7$$\"31+v=Yj*[*=Ffr$\"3kF?lG'fq*oF\\u7$$\"3'* \\iN1j_**=Ffr$\"3=%f@&**[s_vF\\u7$$\"31]PH+nb.>Ffr$\"3SYCl))\\QR\")F\\ u7$$\"3'**\\ihj4z!>Ffr$\"33iFXwNG%y)F\\u7$$\"3-++&[k1C\">Ffr$\"3i^TtDv ic%*F\\u7$$\"33+D^!41o\">Ffr$\"32[$*>bwQ65!#:7$$\"36]iS!yi5#>Ffr$\"3$e `;j]DV2\"Ff[l7$$\"3-+vy3\"*yD>Ffr$\"3'\\^z*\\BxU6Ff[l7$$\"31++0Df.I>Ff r$\"3=%=H4[JC?\"Ff[l7$$\"3-+](yIqX$>Ffr$\"3/RO0>Qoj7Ff[l7$$\"3)*\\ilK \"z'Q>Ffr$\"3jW,.gbj;8Ff[l7$$\"3.+]Uj6Ffr$\"3zk'fD4F9P\"Ff[l7$$\"3) *\\iI9yRZ>Ffr$\"3mI(4f]4)>9Ff[l7$$\"37]7L#)e\"=&>Ffr$\"3\"*3`9\"Q^pY\" Ff[l7$$\"3.+]P^`8c>Ffr$\"3S%4\"Hh#)e4:Ff[l7$$\"3!*\\i:Nulg>Ffr$\"3+Y) \\B&ed]:Ff[l7$$\"3-++IRF,l>Ffr$\"3S$HW(*H$f'e\"Ff[l7$$\"3/+v$omm%p>Ffr $\"3*e;L+SS+i\"Ff[l7$$\"3,]ig8P)Q(>Ffr$\"3#4\\zrx!**\\;Ff[l7$$\"3%**** \\3_Uz(>Ffr$\"3^;5]#HZ[n\"Ff[l7$$\"3$**\\()>P%f#)>Ffr$\"3OE?#***)3/q\" Ff[l7$$\"3(*****4V]v')>Ffr$\"3$>o$H*\\>3s\"Ff[l7$$\"33]iI'=\">\"*>Ffr$ \"3Wn49GNCSFfr$\"3[?<+T6xc