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}{TEXT 252 0 "" }} {PARA 239 "" 0 "" {TEXT 249 0 "" }}{PARA 243 "" 0 "" {TEXT 253 91 "A n onlinear first order diferential equation with singularities at y \+ = + - 2*sqrt(3)/3 " }{TEXT 253 0 "" }}{PARA 239 "" 0 "" {TEXT 249 0 "" }}}{EXCHG {PARA 241 "> " 0 "" {MPLTEXT 1 251 19 "evalf(2*sqrt(3)/3);" }{MPLTEXT 1 251 0 "" }}}{EXCHG {PARA 240 "> " 0 "" {MPLTEXT 1 250 38 "eq22:=diff(y(x),x)=3*x^2/(3*y(x)^2-4);" }{MPLTEXT 1 250 0 "" }}} {EXCHG {PARA 240 "> " 0 "" {MPLTEXT 1 250 95 "DEplot(eq22, y(x),x=-3.. 3, y=-3..3, dirgrid=[30,30],stepsize=0.1,arrows=THIN,title=`Figure 3`) ;" }{MPLTEXT 1 250 0 "" }}}{EXCHG {PARA 244 "" 0 "" {TEXT 254 155 "Thi s graph exhibits an interesting behavior for values of y(x) greater th an y = 2*sqrt(3)/3 and less than y = - 2*sqrt(3)/3, the solutions se ems to grow." }{TEXT 254 0 "" }}{PARA 244 "" 0 "" {TEXT 254 61 "For va lues of y(x) in between these two vlaues they decrease." }{TEXT 254 0 "" }}{PARA 245 "" 0 "" {TEXT 255 0 "" }}}{EXCHG {PARA 240 "> " 0 "" {MPLTEXT 1 250 90 "DEplot(eq22, y(x),x=-3..3, y=-3..3, dirgrid=[30,30] ,\{[1,0]\},arrows=THIN,title=`Figure 3`);" }{MPLTEXT 1 250 0 "" }}} {EXCHG {PARA 244 "" 0 "" {TEXT 254 98 "Poor approximations. A way to i mprove this is to reduce the stepsize in the numerical computation." } {TEXT 254 0 "" }}{PARA 246 "" 0 "" {TEXT 256 19 "The option stepsize" }{TEXT 256 106 " is concerned with the stepsize of the numerical schem e used. The default for an interval a..b is (b-a)/20" }{TEXT 256 0 "" }}{PARA 247 "" 0 "" {TEXT 257 177 "Therefore, in our case stepsize=6/2 0. A smaller stepsize corresponds to a grid refinement of the numerica l scheme in use. As a consequence the solution curve will look smoothe r." }{TEXT 257 0 "" }}{PARA 245 "" 0 "" {TEXT 255 0 "" }}}{EXCHG {PARA 240 "> " 0 "" {MPLTEXT 1 250 104 "DEplot(eq22, y(x),x=-3..3, y=- 3..3, dirgrid=[30,30],\{[1,0]\},stepsize=0.01,arrows=THIN,title=`Figur e 3`);" }{MPLTEXT 1 250 0 "" }}}{EXCHG {PARA 244 "" 0 "" {TEXT 254 70 "We can see now where the limits of the interval of validity should be ." }{TEXT 254 0 "" }}{PARA 244 "" 0 "" {TEXT 254 0 "" }}{PARA 244 "" 0 "" {TEXT 254 39 "Exact solution by hand for I.C. y(1)=0." }{TEXT 254 0 "" }}{PARA 244 "" 0 "" {TEXT 254 0 "" }}}{EXCHG {PARA 241 "> " 0 "" {MPLTEXT 1 251 31 "solneq22:=(x,y)->y^3-4*y-x^3+1;" }{MPLTEXT 1 251 0 "" }}}{EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 -1 62 "implicitplot(solneq 22(x,y)=0,x=-2..2,y=-5..5,numpoints=10000);" }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 241 "> " 0 "" {MPLTEXT 1 251 34 "fsolve(solneq22(x,2*sqrt (3)/3),x);" }{MPLTEXT 1 251 0 "" }}}{EXCHG {PARA 241 "> " 0 "" {MPLTEXT 1 251 35 "fsolve(solneq22(x,-2*sqrt(3)/3),x);" }{MPLTEXT 1 251 0 "" }}}{EXCHG {PARA 240 "> " 0 "" {MPLTEXT 1 250 108 "DEplot(eq22 , y(x),x=-1.2..1.5, y=-3..3, dirgrid=[30,30],\{[1,0]\},stepsize=0.01,a rrows=THIN,title=`Figure 3`);" }{MPLTEXT 1 250 0 "" }}}{EXCHG {PARA 244 "" 0 "" {TEXT 254 64 "Considering other initial values. Explore be havior of solutions." }{TEXT 254 0 "" }}{PARA 245 "" 0 "" {TEXT 255 0 "" }}}{EXCHG {PARA 240 "> " 0 "" {MPLTEXT 1 250 112 "DEplot(eq22, y(x) ,x=-3..1.5, y=-1..1, dirgrid=[30,30],stepsize=0.1,\{[-1,1],[1,1]\},arr ows=THIN,title=`Figure 3`);" }{MPLTEXT 1 250 0 "" }}}{EXCHG {PARA 240 "> " 0 "" {MPLTEXT 1 250 118 "DEplot(eq22, y(x),x=-3..3, y=-3..3, dirg rid=[30,30],stepsize=0.001,\{[-2,1],[2,2],[0,2]\},arrows=THIN,title=`F igure 3`);" }{MPLTEXT 1 250 0 "" }}}{EXCHG {PARA 240 "> " 0 "" {MPLTEXT 1 250 112 "DEplot(eq22, y(x),x=-1..3, y=-3..3, dirgrid=[30,30 ],stepsize=0.001,\{[0,2],[0,-2]\},arrows=THIN,title=`Figure 3`);" } {MPLTEXT 1 250 0 "" }}}{EXCHG {PARA 244 "" 0 "" {TEXT 254 42 "Trying M APLE to obtain the exact solution." }{TEXT 254 0 "" }}{PARA 245 "" 0 " " {TEXT 255 0 "" }}}{EXCHG {PARA 241 "> " 0 "" {MPLTEXT 1 251 18 "dsol ve(eq22,y(x));" }{MPLTEXT 1 251 0 "" }}}{EXCHG {PARA 244 "" 0 "" {TEXT 254 10 "Too messy!" }{TEXT 254 0 "" }}}{EXCHG {PARA 245 "" 0 "" {TEXT 255 0 "" }}{PARA 244 "" 0 "" {TEXT 254 88 "We can also graph the integral curve directly from the hand solution using implicitplot." } {TEXT 254 0 "" }}{PARA 245 "" 0 "" {TEXT 255 0 "" }}}{EXCHG {PARA 240 "> " 0 "" {MPLTEXT 1 250 65 "implicitplot(\{solneq22(x,y)=0\}, x=-2..2 ,y=-5..5,numpoints=10000);" }{MPLTEXT 1 250 0 "" }}}{EXCHG {PARA 244 " " 0 "" {TEXT 254 108 "The option numpoints is the number of points tha t will be use in the function evaluation. The default is 50." }{TEXT 254 0 "" }}{PARA 244 "" 0 "" {TEXT 254 0 "" }}{PARA 244 "" 0 "" {TEXT 254 16 "INTEGRAL CURVES:" }{TEXT 254 0 "" }}{PARA 244 "" 0 "" {TEXT 254 0 "" }}}{EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 0 36 "solneq22gen:=( x,y,c)->y^3-4*y-x^3+c;" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 249 "" 0 "" {TEXT 228 0 "" }}{PARA 249 "" 0 "" {TEXT 228 167 "Obtaining the const ant \"c\" for different initial conditions and graphing the correspond ing integral curves in the xy-plane. using the MAPLE subroutine \"imp licitplot\"." }{TEXT 228 0 "" }}{PARA 249 "" 0 "" {TEXT 228 0 "" }}} {EXCHG {PARA 248 "> " 0 "" {MPLTEXT 1 0 35 "c1:=fsolve(solneq22gen(0,0 ,c)=0,c);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 250 "> " 0 "" {MPLTEXT 1 -1 37 "c2:=fsolve(solneq22gen(0,1/2,c)=0,c);" }{MPLTEXT 1 -1 0 "" }} }{EXCHG {PARA 250 "> " 0 "" {MPLTEXT 1 -1 38 "c3:=fsolve(solneq22gen(0 ,-1/2,c)=0,c);" }{MPLTEXT 1 -1 0 "" }}}{EXCHG {PARA 250 "> " 0 "" {MPLTEXT 1 -1 35 "c4:=fsolve(solneq22gen(0,3,c)=0,c);" }{MPLTEXT 1 -1 0 "" }}}{EXCHG {PARA 250 "> " 0 "" {MPLTEXT 1 -1 36 "c5:=fsolve(solneq 22gen(0,-3,c)=0,c);" }{MPLTEXT 1 -1 0 "" }}}{EXCHG {PARA 250 "> " 0 "" {MPLTEXT 1 -1 35 "c6:=fsolve(solneq22gen(2,0,c)=0,c);" }{MPLTEXT 1 -1 0 "" }}}{EXCHG {PARA 241 "> " 0 "" {MPLTEXT 1 251 35 "implicitplot( \{solneq22gen(x,y,0)=0," }{MPLTEXT 1 251 131 "solneq22gen(x,y,c1)=0,so lneq22gen(x,y,c2)=0,solneq22gen(x,y,c3)=0,solneq22gen(x,y,c4)=0,solneq 22gen(x,y,c5)=0,solneq22gen(x,y,c6)=0" }{MPLTEXT 1 251 36 "\}, x=-5..5 ,y=-5..5,numpoints=10000);" }{MPLTEXT 1 251 0 "" }}}{EXCHG {PARA 241 " > " 0 "" {MPLTEXT 1 251 0 "" }}}{PARA 251 "" 0 "" {TEXT 221 0 "" }} {PARA 252 "" 0 "" {TEXT 210 0 "" }}{PARA 253 "" 0 "" {TEXT 258 0 "" }} {PARA 254 "" 0 "" {TEXT -1 0 "" }}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 15 10 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }