{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "" -1 256 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 257 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 256 "" 0 "" {TEXT 257 90 "Calculus Exploration 15A: Visu alizing Slope Fields for First-Order Differential Equations " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 445 "In each exerc ise below, (1) view the slope field for the given derivative (differen tial) equation and observe how each specific example solution curve \" fits into the arrows\" that represent the tangent lines to the solutio n curve, (2) derive the general solution displayed, yourself by hand ( no calculator), and (3) using each of the given initial conditions, fi nd each corresponding specific solution displayed, yourself by hand (n o calculator)." }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 258 200 "Finally, after completing Exercises 1.1 -1.11 below, create two new interesting differential equations and slo pe fields yourself, using (appropriately editing) the two sections at \+ the end of this file." }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 24 "Slope Field Exercise 1.1" }} {EXCHG {PARA 0 "" 0 "" {TEXT 259 40 "Click in the red area and press [ Enter]." }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1493 "restart :\nwith(DEtools):\nDf:=D(f):\ndiffeq:=[Df(x)=0]:\nx0:=0:\nn:=4:\n##### #############\n## Create Lists ##\n##################\ninitcondsList:= []:\nsolnsList:=[]:\nprintList:=[]:\nfor i from 0 to n do\ninitcondsLi st:=[op(initcondsList),[f(x0)= -2 + i]]:\nsolnsList:=[op(solnsList),[d solve([op(1,diffeq),op(1,op(i+1,initcondsList))])]]:\nprintList:=[op(p rintList),`DiffEq `=op(1,diffeq),`InitCond `=op(1,op(i+1,initcondsList ))]:\nod:\n######################\n## Begin plot ouput ##\n########### ###########\nDEplot(diffeq,[f(x)],\n x=-2..2,f(x)=-3..3,initcond sList, \n linecolour=blue,arrows=MEDIUM,color=red,\n \+ #view=[-1..1,0..1],\n #tickmarks=[4,6],\n labels=[` `,` `] ,\n titlefont=[HELVETICA,DEFAULT,14],\n title=`Slope Field Arrows indicate Multiple Possible Solutions\\n Blue Curves are Exampl es of Specific Solutions`);\n##########################\n## Begin form ula output ##\n##########################\n`Derivative (Differential) \+ Equation: `*`f '`(x)=0,'Df'(x)=0,'dy/dx'=0;\n`Find a function(s) whose tangent lines all have this slope relationship.`;\n`Mutiple Solution \+ Functions: `*Int(`f '`(x),x)=Int(0,x)*`= C`;\n `So, `*'f'(x)=C,`and he re are some Example Solutions: `;\nprint(op(solnsList));\n`Specificati on of a unique solution requires a specific initial condition:`;\nfor \+ i from 0 to n do\n#print[op(solnsList),dsolve([op(1,diffeq),op(1,op(i+ 1,initcondsList))])]:\nprint(`DiffEq `=[`f '`(x)=0],`InitCond `=op(n+1 -i,initcondsList),`Solution `=op(n+1-i,solnsList));\nod:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 24 "Slope Field Exercise 1.2" }}{EXCHG {PARA 0 "" 0 "" {TEXT 260 40 "Click in the red area and press [Enter]." }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1598 "restart:\nwith(DEtools):\nDf:=D(f):\ng: =x->1:\ndiffeq:=[Df(x)=g(x)]:\ngensoln:=dsolve(op(1,diffeq)):\n_C1:=0: \nh:=unapply(rhs(gensoln),x):\nx0:=0:\nn:=4:\n##################\n## C reate Lists ##\n##################\ninitcondsList:=[]:\nsolnsList:=[]: \nprintList:=[]:\nfor i from 0 to n do\ninitcondsList:=[op(initcondsLi st),[f(x0)= -2 + i]]:\nsolnsList:=[op(solnsList),[dsolve([op(1,diffeq) ,op(1,op(i+1,initcondsList))])]]:\nprintList:=[op(printList),`DiffEq ` =op(1,diffeq),`InitCond `=op(1,op(i+1,initcondsList))]:\nod:\n######## ##############\n## Begin plot ouput ##\n######################\nDEplot (diffeq,[f(x)],\n x=-2..2,f(x)=-3..3,initcondsList, \n \+ linecolour=blue,arrows=MEDIUM,color=red,\n #view=[-1..1,0..1], \n #tickmarks=[4,6],\n labels=[` `,` `],\n titlefont =[HELVETICA,DEFAULT,14],\n title=`Slope Field Arrows indicate Mu ltiple Possible Solutions\\n Blue Curves are Examples of Specific Solu tions`);\n##########################\n## Begin formula output ##\n#### ######################\n`Derivative (Differential) Equation: `*`f '`(x )=g(x),'Df'(x)=g(x),'dy/dx'=g(x);\n`Find a function(s) whose tangent l ines all have this slope relationship.`;\n`Mutiple Solution Functions: `*Int(`f '`(x),x)=Int(g(x),x)*`= `*h(x)+C;\n `So, `*'f'(x)=h(x)+C,`an d here are some Example Solutions: `;\nprint(op(solnsList));\n`Specifi cation of a unique solution requires a specific initial condition:`;\n for i from 0 to n do\n#print[op(solnsList),dsolve([op(1,diffeq),op(1,o p(i+1,initcondsList))])]:\nprint(`DiffEq `=[`f '`(x)=g(x)],`InitCond ` =op(n+1-i,initcondsList),`Solution `=op(n+1-i,solnsList));\nod:\n" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 " " {TEXT -1 24 "Slope Field Exercise 1.3" }}{EXCHG {PARA 0 "" 0 "" {TEXT 261 40 "Click in the red area and press [Enter]." }{TEXT -1 0 " " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1599 "restart:\nwith(DEtools):\nDf: =D(f):\ng:=x->x:\ndiffeq:=[Df(x)=g(x)]:\ngensoln:=dsolve(op(1,diffeq)) :\n_C1:=0:\nh:=unapply(rhs(gensoln),x):\nx0:=0:\nn:=4:\n############## ####\n## Create Lists ##\n##################\ninitcondsList:=[]:\nsoln sList:=[]:\nprintList:=[]:\nfor i from 0 to n do\ninitcondsList:=[op(i nitcondsList),[f(x0)= -2 + i]]:\nsolnsList:=[op(solnsList),[dsolve([op (1,diffeq),op(1,op(i+1,initcondsList))])]]:\nprintList:=[op(printList) ,`DiffEq `=op(1,diffeq),`InitCond `=op(1,op(i+1,initcondsList))]:\nod: \n######################\n## Begin plot ouput ##\n#################### ##\nDEplot(diffeq,[f(x)],\n x=-2..2,f(x)=-3..3,initcondsList, \+ \n linecolour=blue,arrows=MEDIUM,color=red,\n #view=[-1 ..1,0..1],\n #tickmarks=[4,6],\n labels=[` `,` `],\n \+ titlefont=[HELVETICA,DEFAULT,14],\n title=`Slope Field Arrows i ndicate Multiple Possible Solutions\\n Blue Curves are Examples of Spe cific Solutions`);\n##########################\n## Begin formula outpu t ##\n##########################\n`Derivative (Differential) Equation: `*`f '`(x)=g(x),'Df'(x)=g(x),'dy/dx'=g(x);\n`Find a function(s) whose tangent lines all have this slope relationship.`;\n`Mutiple Solution \+ Functions: `*Int(`f '`(x),x)*` = `*Int(g(x),x)=h(x)+C;\n `So, `*'f'(x) =h(x)+C,`and here are some Example Solutions: `;\nprint(op(solnsList)) ;\n`Specification of a unique solution requires a specific initial con dition:`;\nfor i from 0 to n do\n#print[op(solnsList),dsolve([op(1,dif feq),op(1,op(i+1,initcondsList))])]:\nprint(`DiffEq `=[`f '`(x)=g(x)], `InitCond `=op(n+1-i,initcondsList),`Solution `=op(n+1-i,solnsList)); \nod:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 24 "Slope Field Exercise 1.4" }}{EXCHG {PARA 0 "" 0 "" {TEXT 262 40 "Click in the red area and press [Enter]." } {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1600 "restart:\nwith(DE tools):\nDf:=D(f):\ng:=x->x^2:\ndiffeq:=[Df(x)=g(x)]:\ngensoln:=dsolve (op(1,diffeq)):\n_C1:=0:\nh:=unapply(rhs(gensoln),x):\nx0:=0:\nn:=4:\n ##################\n## Create Lists ##\n##################\ninitcondsL ist:=[]:\nsolnsList:=[]:\nprintList:=[]:\nfor i from 0 to n do\ninitco ndsList:=[op(initcondsList),[f(x0)= -2 + i]]:\nsolnsList:=[op(solnsLis t),[dsolve([op(1,diffeq),op(1,op(i+1,initcondsList))])]]:\nprintList:= [op(printList),`DiffEq `=op(1,diffeq),`InitCond `=op(1,op(i+1,initcond sList))]:\nod:\n######################\n## Begin plot ouput ##\n###### ################\nDEplot(diffeq,[f(x)],\n x=-2..2,f(x)=-3..3,ini tcondsList, \n linecolour=blue,arrows=MEDIUM,color=red,\n \+ #view=[-1..1,0..1],\n #tickmarks=[4,6],\n labels=[` ` ,` `],\n titlefont=[HELVETICA,DEFAULT,14],\n title=`Slope \+ Field Arrows indicate Multiple Possible Solutions\\n Blue Curves are E xamples of Specific Solutions`);\n##########################\n## Begin formula output ##\n##########################\n`Derivative (Different ial) Equation: `*`f '`(x)=g(x),'Df'(x)=g(x),'dy/dx'=g(x);\n`Find a fun ction(s) whose tangent lines all have this slope relationship.`;\n`Mut iple Solution Functions: `*Int(`f '`(x),x)*` = `*Int(g(x),x)=h(x)+C;\n `So, `*'f'(x)=h(x)+C,`and here are some Example Solutions: `;\nprint( op(solnsList));\n`Specification of a unique solution requires a specif ic initial condition:`;\nfor i from 0 to n do\n#print[op(solnsList),ds olve([op(1,diffeq),op(1,op(i+1,initcondsList))])]:\nprint(`DiffEq `=[` f '`(x)=g(x)],`InitCond `=op(n+1-i,initcondsList),`Solution `=op(n+1-i ,solnsList));\nod:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 24 "Slope Field Exercise 1.5" }} {EXCHG {PARA 0 "" 0 "" {TEXT 263 40 "Click in the red area and press [ Enter]." }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1604 "restart :\nwith(DEtools):\nDf:=D(f):\ng:=x->cos(x):\ndiffeq:=[Df(x)=g(x)]:\nge nsoln:=dsolve(op(1,diffeq)):\n_C1:=0:\nh:=unapply(rhs(gensoln),x):\nx0 :=0:\nn:=4:\n##################\n## Create Lists ##\n################# #\ninitcondsList:=[]:\nsolnsList:=[]:\nprintList:=[]:\nfor i from 0 to n do\ninitcondsList:=[op(initcondsList),[f(x0)= -2 + i]]:\nsolnsList: =[op(solnsList),[dsolve([op(1,diffeq),op(1,op(i+1,initcondsList))])]]: \nprintList:=[op(printList),`DiffEq `=op(1,diffeq),`InitCond `=op(1,op (i+1,initcondsList))]:\nod:\n######################\n## Begin plot oup ut ##\n######################\nDEplot(diffeq,[f(x)],\n x=-2..2,f (x)=-3..3,initcondsList, \n linecolour=blue,arrows=MEDIUM,c olor=red,\n #view=[-1..1,0..1],\n #tickmarks=[4,6],\n \+ labels=[` `,` `],\n titlefont=[HELVETICA,DEFAULT,14],\n \+ title=`Slope Field Arrows indicate Multiple Possible Solutions\\n Blue Curves are Examples of Specific Solutions`);\n####################### ###\n## Begin formula output ##\n##########################\n`Derivati ve (Differential) Equation: `*`f '`(x)=g(x),'Df'(x)=g(x),'dy/dx'=g(x); \n`Find a function(s) whose tangent lines all have this slope relation ship.`;\n`Mutiple Solution Functions: `*Int(`f '`(x),x)*` = `*Int(g(x) ,x)=h(x)+C;\n `So, `*'f'(x)=h(x)+C,`and here are some Example Solution s: `;\nprint(op(solnsList));\n`Specification of a unique solution requ ires a specific initial condition:`;\nfor i from 0 to n do\n#print[op( solnsList),dsolve([op(1,diffeq),op(1,op(i+1,initcondsList))])]:\nprint (`DiffEq `=[`f '`(x)=g(x)],`InitCond `=op(n+1-i,initcondsList),`Soluti on `=op(n+1-i,solnsList));\nod:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 24 "Slope Field Exercise 1.6" }}{EXCHG {PARA 0 "" 0 "" {TEXT 264 40 "Click in the red area and press [Enter]." }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1769 "restart:\nwith(DEtools):\nDf:=D(f):\ng:=x->exp(x): # Enter \+ function on the right of the arrow\nx0:=0: # Enter base value \+ of x0 for initial conditions\nn:=4: # Enter the number n of a dditional initial conditions\ndiffeq:=[Df(x)=g(x)]:\ngensoln:=dsolve(o p(1,diffeq)):\nh:=unapply(rhs(gensoln),x):\n##################\n## Cre ate Lists ##\n##################\ninitcondsList:=[]:\nsolnsList:=[]:\n printList:=[]:\nfor i from 0 to n do\ninitcondsList:=[op(initcondsList ),[f(x0)= -2 + i]]:\nsolnsList:=[op(solnsList),[dsolve([op(1,diffeq),o p(1,op(i+1,initcondsList))])]]:\nprintList:=[op(printList),`DiffEq `=o p(1,diffeq),`InitCond `=op(1,op(i+1,initcondsList))]:\nod:\n########## ############\n## Begin plot ouput ##\n######################\nDEplot(d iffeq,[f(x)],\n x=-2..2,f(x)=-3..3,initcondsList, \n \+ linecolour=blue,arrows=MEDIUM,color=red,\n #view=[-1..1,0..1],\n #tickmarks=[4,6],\n labels=[` `,` `],\n titlefont=[ HELVETICA,DEFAULT,14],\n title=`Slope Field Arrows indicate Mult iple Possible Solutions\\n Blue Curves are Examples of Specific Soluti ons`);\n##########################\n## Begin formula output ##\n###### ####################\n`Derivative (Differential) Equation: `*`f '`(x)= g(x),'Df'(x)=g(x),'dy/dx'=g(x);\n`Find a function(s) whose tangent lin es all have this slope relationship.`;\n`Mutiple Solution Functions: ` *Int(`f '`(x),x)*` = `*Int(g(x),x)=h(x)+C -_C1;\n `So, `*'f'(x)=h(x)+C -_C1,`and here are some Example Solutions: `;\nprint(op(solnsList)); \n`Specification of a unique solution requires a specific initial cond ition:`;\nfor i from 0 to n do\n#print[op(solnsList),dsolve([op(1,diff eq),op(1,op(i+1,initcondsList))])]:\nprint(`DiffEq `=[`f '`(x)=g(x)],` InitCond `=op(n+1-i,initcondsList),`Solution `=op(n+1-i,solnsList));\n od:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 24 "Slope Field Exercise 1.7" }}{EXCHG {PARA 0 "" 0 "" {TEXT 265 40 "Click in the red area and press [Enter]." } {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1826 "restart:\nwith(DE tools):\nDf:=D(f):\ng:=x->f(x): # Enter function on the right of the \+ arrow\nx0:=0: # Enter base value of x0 for initial conditions\nn :=4: # Enter the number n of additional initial conditions\ndif feq:=[Df(x)=g(x)]:\ngensoln:=dsolve(op(1,diffeq)):\nh:=unapply(rhs(gen soln),x):\n##################\n## Create Lists ##\n################## \ninitcondsList:=[]:\nsolnsList:=[]:\nprintList:=[]:\nfor i from 0 to \+ n do\ninitcondsList:=[op(initcondsList),[f(x0)= -2 + i]]:\nsolnsList:= [op(solnsList),[dsolve([op(1,diffeq),op(1,op(i+1,initcondsList))])]]: \nprintList:=[op(printList),`DiffEq `=op(1,diffeq),`InitCond `=op(1,op (i+1,initcondsList))]:\nod:\n######################\n## Begin plot oup ut ##\n######################\nDEplot(diffeq,[f(x)],\n x=-2..2,f (x)=-3..3,initcondsList, \n linecolour=blue,arrows=MEDIUM,c olor=red,\n #view=[-1..1,0..1],\n #tickmarks=[4,6],\n \+ labels=[` `,` `],\n titlefont=[HELVETICA,DEFAULT,14],\n \+ title=`Slope Field Arrows indicate Multiple Possible Solutions\\n Blue Curves are Examples of Specific Solutions`);\n####################### ###\n## Begin formula output ##\n##########################\n`Derivati ve (Differential) Equation: `*`f '`(x)=g(x),'Df'(x)=g(x),'dy/dx'=y;\n` Find a function(s) whose tangent lines all have this slope relationshi p.`;\n`Mutiple Solution Functions: `;\nInt(`f '`(x)/f(x),x)=Int(1,x); \nln(f(x)) = x+C;\nexp(ln(f(x)))*` = `*exp(x+C)=exp(x)*exp(C);\n\n `So , `*'f'(x)=(K/_C1)*h(x),` where `*K=exp(C),`and here are some Example \+ Solutions: `;\nprint(op(solnsList));\n`Specification of a unique solut ion requires a specific initial condition:`;\nfor i from 0 to n do\n#p rint[op(solnsList),dsolve([op(1,diffeq),op(1,op(i+1,initcondsList))])] :\nprint(`DiffEq `=[`f '`(x)=g(x)],`InitCond `=op(n+1-i,initcondsList) ,`Solution `=op(n+1-i,solnsList));\nod:\n" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 24 "Slope Fie ld Exercise 1.8" }}{EXCHG {PARA 0 "" 0 "" {TEXT 266 40 "Click in the r ed area and press [Enter]." }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1738 "restart:\nwith(DEtools):\nDf:=D(f):\ng:=x->x+f(x): \+ # Enter function on the right of the arrow\nx0:=0: # Enter base value of x0 for initial conditions\nn:=4: # Enter the number n of additional initial conditions\ndiffeq:=[Df(x)=g(x)]:\ngensoln:=dso lve(op(1,diffeq)):\nh:=unapply(rhs(gensoln),x):\n##################\n# # Create Lists ##\n##################\ninitcondsList:=[]:\nsolnsList:= []:\nprintList:=[]:\nfor i from 0 to n do\ninitcondsList:=[op(initcond sList),[f(x0)= -2 + i]]:\nsolnsList:=[op(solnsList),[dsolve([op(1,diff eq),op(1,op(i+1,initcondsList))])]]:\nprintList:=[op(printList),`DiffE q `=op(1,diffeq),`InitCond `=op(1,op(i+1,initcondsList))]:\nod:\n##### #################\n## Begin plot ouput ##\n######################\nDEp lot(diffeq,[f(x)],\n x=-2..2,f(x)=-3..3,initcondsList, \n \+ linecolour=blue,arrows=MEDIUM,color=red,\n #view=[-1..1,0.. 1],\n #tickmarks=[4,6],\n labels=[` `,` `],\n titlef ont=[HELVETICA,DEFAULT,14],\n title=`Slope Field Arrows indicate Multiple Possible Solutions\\n Blue Curves are Examples of Specific S olutions`);\n##########################\n## Begin formula output ##\n# #########################\n`Derivative (Differential) Equation: `*`f ' `(x)=g(x),'Df'(x)=g(x),'dy/dx'=subs(f(x)=y,g(x));\n`Find all functions whose tangent lines all have this slope relationship.`;\n\n`Mutiple S olution Functions: `*'f'(x)=subs(exp(x)*_C1=K*exp(x),h(x)),` where K i s an arbitrary constant.`;\n`Example Solutions: Specification of a uni que solution requires a specific initial condition.`;\nfor i from 0 to n do\n#print[op(solnsList),dsolve([op(1,diffeq),op(1,op(i+1,initconds List))])]:\nprint(`DiffEq `=[`f '`(x)=g(x)],`InitCond `=op(n+1-i,initc ondsList),`Solution `=op(n+1-i,solnsList));\nod:\n\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 24 "Slope Field Exercise 1.9" }}{EXCHG {PARA 0 "" 0 "" {TEXT 267 40 "Clic k in the red area and press [Enter]." }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1722 "restart:\nwith(DEtools):\nDf:=D(f):\ng:=x->x*f(x ): # Enter function on the right of the arrow\nx0:=0: # Enter b ase value of x0 for initial conditions\nn:=4: # Enter the numbe r n of additional initial conditions\ndiffeq:=[Df(x)=g(x)]:\ngensoln:= dsolve(op(1,diffeq)):\nh:=unapply(rhs(gensoln),x):\n################## \n## Create Lists ##\n##################\ninitcondsList:=[]:\nsolnsLis t:=[]:\nprintList:=[]:\nfor i from 0 to n do\ninitcondsList:=[op(initc ondsList),[f(x0)= -2 + i]]:\nsolnsList:=[op(solnsList),[dsolve([op(1,d iffeq),op(1,op(i+1,initcondsList))])]]:\nprintList:=[op(printList),`Di ffEq `=op(1,diffeq),`InitCond `=op(1,op(i+1,initcondsList))]:\nod:\n## ####################\n## Begin plot ouput ##\n######################\n DEplot(diffeq,[f(x)],\n x=-2..2,f(x)=-3..3,initcondsList, \+ \n linecolour=blue,arrows=MEDIUM,color=red,\n #view=[-1..1 ,0..1],\n #tickmarks=[4,6],\n labels=[` `,` `],\n ti tlefont=[HELVETICA,DEFAULT,14],\n title=`Slope Field Arrows indi cate Multiple Possible Solutions\\n Blue Curves are Examples of Specif ic Solutions`);\n##########################\n## Begin formula output # #\n##########################\n`Derivative (Differential) Equation: `* `f '`(x)=g(x),'Df'(x)=g(x),'dy/dx'=subs(f(x)=y,g(x));\n`Find all funct ions whose tangent lines all have this slope relationship.`;\n\n`Mutip le Solution Functions: `*'f'(x)=subs(_C1=K,h(x)),` where K is an arbit rary constant.`;\n`Example Solutions: Specification of a unique soluti on requires a specific initial condition.`;\nfor i from 0 to n do\n#pr int[op(solnsList),dsolve([op(1,diffeq),op(1,op(i+1,initcondsList))])]: \nprint(`DiffEq `=[`f '`(x)=g(x)],`InitCond `=op(n+1-i,initcondsList), `Solution `=op(n+1-i,solnsList));\nod:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 25 "Slope Field Exercise 1.10" }}{EXCHG {PARA 0 "" 0 "" {TEXT 268 40 "Click in the re d area and press [Enter]." }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1965 "restart:\nwith(DEtools):\nDf:=D(f):\ng:=x->-f(x)/x: # Enter function on the right of the arrow\nx0:=1: # Enter \+ base value of x0 for initial conditions\nn:=4: # Enter the n umber n of additional initial conditions\nxleft:=-2: # Enter min \+ value of x for the plot\nxright:=2: # Enter max value of x for th e plot\nybot:=-3: # Enter min value of y for the plot\nytop:=3: \+ # Enter max value of y for the plot \ndiffeq:=[D f(x)=g(x)]:\ngensoln:=dsolve(op(1,diffeq)):\nh:=unapply(rhs(gensoln),x ):\n##################\n## Create Lists ##\n##################\ninitco ndsList:=[]:\nsolnsList:=[]:\nprintList:=[]:\nfor i from 0 to n do\nin itcondsList:=[op(initcondsList),[f(x0)= -2 + i]]:\nsolnsList:=[op(soln sList),[dsolve([op(1,diffeq),op(1,op(i+1,initcondsList))])]]:\nprintLi st:=[op(printList),`DiffEq `=op(1,diffeq),`InitCond `=op(1,op(i+1,init condsList))]:\nod:\n######################\n## Begin plot ouput ##\n## ####################\nDEplot(diffeq,[f(x)],\n x=xleft..xright,f( x)=ybot..ytop,initcondsList, \n linecolour=blue,arrows=MEDI UM,color=red,\n #view=[-1..1,0..1],\n #tickmarks=[4,6],\n \+ labels=[` `,` `],\n titlefont=[HELVETICA,DEFAULT,14],\n \+ title=`Slope Field Arrows indicate Multiple Possible Solutions\\n \+ Blue Curves are Examples of Specific Solutions`);\n################### #######\n## Begin formula output ##\n##########################\n`Deri vative (Differential) Equation: `*`f '`(x)=g(x),'Df'(x)=g(x),'dy/dx'=s ubs(f(x)=y,g(x));\n`Find all functions whose tangent lines all have th is slope relationship.`;\n`Mutiple Solution Functions: `*'f'(x)=subs(_ C1=K,h(x)),` where K is an arbitrary constant.`;\n`Example Solutions: \+ Specification of a unique solution requires a specific initial conditi on.`;\nfor i from 0 to n do\n#print[op(solnsList),dsolve([op(1,diffeq) ,op(1,op(i+1,initcondsList))])]:\nprint(`DiffEq `=[`f '`(x)=g(x)],`Ini tCond `=op(n+1-i,initcondsList),`Solution `=op(n+1-i,solnsList));\nod: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 " " 0 "" {TEXT -1 25 "Slope Field Exercise 1.11" }}{EXCHG {PARA 0 "" 0 " " {TEXT 269 40 "Click in the red area and press [Enter]." }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 2036 "restart:\nwith(DEtools):\nDf :=D(f):\ng:=x->-x/f(x): # Enter function on the right of the arrow\nx 0:=0: # Enter base value of x0 for initial conditions\nm:=2: \+ # Enter the number m (radius in this example)\nxleft:=-m: \+ # Enter min value of x for the plot\nxright:=m: # Enter max va lue of x for the plot\nybot:=-m: # Enter min value of y for the \+ plot\nytop:=m: # Enter max value of y for the plot\nn:=1: \+ \ndiffeq:=[Df(x)=g(x)]:\ngensoln:=dsolve(op(1,diffeq)):\nh1: =unapply(rhs(op(1,[gensoln])),x):\nh2:=unapply(rhs(op(2,[gensoln])),x) :\n##################\n## Create Lists ##\n##################\ninitcon dsList:=[]:\nsolnsList:=[]:\nprintList:=[]:\nfor i from 0 to n do\nini tcondsList:=[op(initcondsList),[f(x0)=((-1)^(i+1))*m]]:\nsolnsList:=[o p(solnsList),[dsolve([op(1,diffeq),op(1,op(i+1,initcondsList))])]]:\np rintList:=[op(printList),`DiffEq `=op(1,diffeq),`InitCond `=op(1,op(i+ 1,initcondsList))]:\nod:\n######################\n## Begin plot ouput \+ ##\n######################\nDEplot(diffeq,[f(x)],\n x=xleft..xri ght,f(x)=ybot..ytop,initcondsList, \n linecolour=blue,arrow s=MEDIUM,color=red,\n #view=[-1..1,0..1],\n #tickmarks=[4, 6],\n labels=[` `,` `],\n titlefont=[HELVETICA,DEFAULT,14] ,\n title=`Slope Field Arrows indicate Multiple Possible Solutio ns\\n Blue Curves are Examples of Specific Solutions`);\n############# #############\n## Begin formula output ##\n########################## \n`Derivative (Differential) Equation: `*`f '`(x)=g(x),'Df'(x)=g(x),'d y/dx'=subs(f(x)=y,g(x));\n`Find all functions whose tangent lines all \+ have this slope relationship.`;\n`Mutiple Solution Functions: `*'f'(x) =subs(_C1=K,h1(x)),subs(_C1=K,h2(x)),` where K is an arbitrary constan t.`;\n`Example Solutions: Specification of a unique solution requires \+ a specific initial condition.`;\nfor i from 0 to n do\n#print[op(solns List),dsolve([op(1,diffeq),op(1,op(i+1,initcondsList))])]:\nprint(`Dif fEq `=[`f '`(x)=g(x)],`InitCond `=op(n+1-i,initcondsList),`Solution `= op(n+1-i,solnsList));\nod:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 16 "My Slope Field 1" }} {EXCHG {PARA 0 "" 0 "" {TEXT 270 40 "Click in the red area and press [ Enter]." }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1963 "restart :\nwith(DEtools):\nDf:=D(f):\ng:=x->f(x)^2: # Enter function on the r ight of the arrow\nx0:=0: # Enter base value of x0 for initia l conditions\nn:=4: # Enter the number n of additional intia l conditions\nxleft:=-2: # Enter min value of x for the plot\nxri ght:=2: # Enter max value of x for the plot\nybot:=-3: # En ter min value of y for the plot\nytop:=3: # Enter max value of \+ y for the plot \ndiffeq:=[Df(x)=g(x)]:\ngensoln:=dsolve (op(1,diffeq)):\nh:=unapply(rhs(gensoln),x):\n##################\n## C reate Lists ##\n##################\ninitcondsList:=[]:\nsolnsList:=[]: \nprintList:=[]:\nfor i from 0 to n do\ninitcondsList:=[op(initcondsLi st),[f(x0)= -2 + i]]:\nsolnsList:=[op(solnsList),[dsolve([op(1,diffeq) ,op(1,op(i+1,initcondsList))])]]:\nprintList:=[op(printList),`DiffEq ` =op(1,diffeq),`InitCond `=op(1,op(i+1,initcondsList))]:\nod:\n######## ##############\n## Begin plot ouput ##\n######################\nDEplot (diffeq,[f(x)],\n x=xleft..xright,f(x)=ybot..ytop,initcondsList, \n linecolour=blue,arrows=MEDIUM,color=red,\n #view= [-1..1,0..1],\n #tickmarks=[4,6],\n labels=[` `,` `],\n \+ titlefont=[HELVETICA,DEFAULT,14],\n title=`Slope Field Arrow s indicate Multiple Possible Solutions\\n Blue Curves are Examples of \+ Specific Solutions`);\n##########################\n## Begin formula ou tput ##\n##########################\n`Derivative (Differential) Equati on: `*`f '`(x)=g(x),'Df'(x)=g(x),'dy/dx'=subs(f(x)=y,g(x));\n`Find all functions whose tangent lines all have this slope relationship.`;\n`M utiple Solution Functions: `*'f'(x)=subs(_C1=K,h(x)),` where K is an a rbitrary constant.`;\n`Example Solutions: Specification of a unique so lution requires a specific initial condition.`;\nfor i from 0 to n do \n#print[op(solnsList),dsolve([op(1,diffeq),op(1,op(i+1,initcondsList) )])]:\nprint(`DiffEq `=[`f '`(x)=g(x)],`InitCond `=op(n+1-i,initcondsL ist),`Solution `=op(n+1-i,solnsList));\nod:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 16 "My Slope Field 2" }}{EXCHG {PARA 0 "" 0 "" {TEXT 271 40 "Click in the red area and press [Enter]." }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1965 "restart:\nwith(DEtools):\nDf:=D(f):\ng:=x->x*f(x)^2: # Enter fu nction on the right of the arrow\nx0:=1: # Enter base value o f x0 for initial conditions\nn:=4: # Enter the number n of a dditional intial conditions\nxleft:=-2: # Enter min value of x fo r the plot\nxright:=2: # Enter max value of x for the plot\nybot: =-3: # Enter min value of y for the plot\nytop:=3: # Ente r max value of y for the plot \ndiffeq:=[Df(x)=g(x)]:\n gensoln:=dsolve(op(1,diffeq)):\nh:=unapply(rhs(gensoln),x):\n######### #########\n## Create Lists ##\n##################\ninitcondsList:=[]: \nsolnsList:=[]:\nprintList:=[]:\nfor i from 0 to n do\ninitcondsList: =[op(initcondsList),[f(x0)= -2 + i]]:\nsolnsList:=[op(solnsList),[dsol ve([op(1,diffeq),op(1,op(i+1,initcondsList))])]]:\nprintList:=[op(prin tList),`DiffEq `=op(1,diffeq),`InitCond `=op(1,op(i+1,initcondsList))] :\nod:\n######################\n## Begin plot ouput ##\n############## ########\nDEplot(diffeq,[f(x)],\n x=xleft..xright,f(x)=ybot..yto p,initcondsList, \n linecolour=blue,arrows=MEDIUM,color=red ,\n #view=[-1..1,0..1],\n #tickmarks=[4,6],\n labels =[` `,` `],\n titlefont=[HELVETICA,DEFAULT,14],\n title=`S lope Field Arrows indicate Multiple Possible Solutions\\n Blue Curves \+ are Examples of Specific Solutions`);\n##########################\n## \+ Begin formula output ##\n##########################\n`Derivative (Diff erential) Equation: `*`f '`(x)=g(x),'Df'(x)=g(x),'dy/dx'=subs(f(x)=y,g (x));\n`Find all functions whose tangent lines all have this slope rel ationship.`;\n`Mutiple Solution Functions: `*'f'(x)=subs(_C1=K,h(x)),` where K is an arbitrary constant.`;\n`Example Solutions: Specificatio n of a unique solution requires a specific initial condition.`;\nfor i from 0 to n do\n#print[op(solnsList),dsolve([op(1,diffeq),op(1,op(i+1 ,initcondsList))])]:\nprint(`DiffEq `=[`f '`(x)=g(x)],`InitCond `=op(n +1-i,initcondsList),`Solution `=op(n+1-i,solnsList));\nod:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "_____________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "Filenam e: ExploreCalc15A.mws" }}{PARA 0 "" 0 "" {TEXT -1 36 "Copyright 2006, \+ All Rights Reserved." }}{PARA 0 "" 0 "" {TEXT -1 44 "Permission is gra nted to use and modify for " }}{PARA 0 "" 0 "" {TEXT -1 37 "academic a nd non-commercial purposes." }}{PARA 0 "" 0 "" {TEXT -1 13 "Dr. John P ais" }}{PARA 0 "" 0 "" {TEXT -1 28 "Mathematics Department-MICDS" }} {PARA 0 "" 0 "" {TEXT -1 45 "E-mail: pais@micds.org or pais@kinetigram .com" }}{PARA 0 "" 0 "" {TEXT -1 33 "URL: http://kinetigram.com/micds " }}{PARA 0 "" 0 "" {TEXT -1 37 "_____________________________________ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {MARK "1 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }