{VERSION 3 0 "IBM INTEL NT" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } {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 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "1.1 Exercises. Hints or a nswers." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "1-14 O/P order ind var dep var linear?" }}{PARA 0 " " 0 "" {TEXT -1 69 "1. o 2 t \+ x yes" }}{PARA 0 "" 0 "" {TEXT -1 68 "2. o \+ 2 x y yes" }}{PARA 0 "" 0 "" {TEXT -1 69 "3. o 1 x y \+ no " }}{PARA 0 "" 0 "" {TEXT -1 67 "4. p \+ 2 x,y u yes" }}{PARA 0 "" 0 "" {TEXT -1 68 "5. o 1 t p \+ no" }}{PARA 0 "" 0 "" {TEXT -1 68 "6. o 1 \+ t x no" }}{PARA 0 "" 0 "" {TEXT -1 67 "7 . o 2 x y no" }} {PARA 0 "" 0 "" {TEXT -1 67 "8. o 2 x \+ y no" }}{PARA 0 "" 0 "" {TEXT -1 68 "9. \+ o 2 x y yes" }}{PARA 0 " " 0 "" {TEXT -1 67 "10. o 4 x \+ y yes" }}{PARA 0 "" 0 "" {TEXT -1 66 "11. p \+ 2 r,t N yes" }}{PARA 0 "" 0 "" {TEXT -1 65 "12. o 2 x y \+ no" }}{PARA 0 "" 0 "" {TEXT -1 65 "13. o 2 \+ x y no" }}{PARA 0 "" 0 "" {TEXT -1 66 "1 4. o 1 t y no" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "15." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "diff(T(t),t)=alpha*(M-T(t)); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "dsolve(%,T(t));" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "16." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "diff(A(t),t)=alpha*A(t)^2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "dsolve(%,A(t));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "17. Accel Alison = aa, accel of Kevin = ak." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "dea:=diff(A(t),t,t)=aa;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "dek:=diff(K(t),t,t)=ak;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "inita:=A(0)=0,D(A)(0)=0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "initk:=K(0)=0,D(K)(0)=0;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "sola:=rhs(dsolve(\{dea,inita \},A(t)));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "solk:=rhs(dso lve(\{dek,initk\},K(t)));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "a:=t->aa*t^2/2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "k:=t ->ak*t^2/2;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 89 "Let d be the total distance, t the time for A to finish, and s the time for K to finish. ." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "e1:=d-a(t-4)=d/3;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "e2:=d-k(s-3)=d/4;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "sol:=solve(\{e1,e2\},\{aa,ak \});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "e3:=d=a(t); e4:=d=k (s);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "solve won't do it, so a l ittle hand work is needed. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "ek:=subs(e4,sol[2]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "ea:=subs(e3,sol[1]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "solve(ek,s);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "solve(ea,t );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "12+4*sqrt(6), 12-4*sq rt(6);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "How do you decide the s ign?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "evalf(6*sqrt(3)); e valf(4*sqrt(6));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "%%-%;" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "So Allison wins by " }{XPPEDIT 18 0 "6*sqrt(3)-4*sqrt(6);" "6#,&*&\"\"'\"\"\"-%%sqrtG6#\"\"$F&F&*&\" \"%F&-F(6#\"\"'F&!\"\"" }{TEXT -1 23 " , about 0.594 seconds." }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "1. 2 page 14-" }}{PARA 0 "" 0 "" {TEXT -1 14 "7. Sub it in:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "d7:=diff(y(x),x,x)-diff(y(x),x)-2*y (x)=0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "sol:=exp(2*x)-3*e xp(-x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "subs(y(x)=sol,d7 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "11. Implicit sol!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "d11:=diff(y(x),x)= (exp(-x*y(x))-y( x))/(exp(-x*y(x))+x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "re l:=exp(x*y(x))+y(x)=x-1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "simplify(subs(rel,d11));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "Didn 't work (that isn't how subs works), so we have to get y' first." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "ider:=diff(rel,x);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "d111:=solve(ider,diff(y(x),x ));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "d112:=simplify(d111= rhs(d11));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "d113:=subs(x*y(x)=v,d112);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "simplify(d113,exp);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "d114:=subs(exp(-v)=1/vv,d113);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "d115:=subs(exp(v)=vv,d114);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "simplify(d115);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "It is finally clear they are equal." }} {PARA 0 "" 0 "" {TEXT -1 3 "17." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "d17:=diff(y(x),x)=y(x)*(y(x)-2)/2;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 23 "phi:=x->2/(1-C*exp(x));" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 35 "subs(y(x)=phi(x),d17); simplify(%);" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "Checks. " }}{PARA 0 "" 0 "" {TEXT -1 8 "20. a, b" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "phi:=x->exp(m*x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "solve(diff(phi(x),x,x)+6*diff(phi(x),x)+5*phi(x),m); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "solve(diff(phi(x),x,x,x )+ 3*diff(phi(x),x,x)+2*diff(phi(x),x),m);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "22. a, b" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "d20 :=diff(y(x),x,x)-diff(y(x),x)-2*y(x)=0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "subs(y(x)=c[1]*exp(-x)+c[2]*exp(2*x),d20);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "phi:=x->c[1]*exp(-x)+c[2]*exp(2*x); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "e1:=subs(subs(x=0,phi(x )=2));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "e2:=subs(x=0,diff (phi(x),x)=1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "solve(\{e 1,e2\},\{c[1],c[2]\});" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "1.3 Di rection Fields" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 123 "Maple tools for sketching direction fields in the DEtool s package: two are DEplot and dfieldplot. Here are some examples." } }{PARA 0 "" 0 "" {TEXT -1 31 "Exercises starting page 22: 4." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "restart:with(DEtools);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "DEplot(diff(v(t),t)=1-v(t)^ 3/8, v(t), t=0..3, [[v(0)=0], [v(0)=1], [v(0)=2], [v(0)=3]], v=0..3,st epsize=.05);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 121 "Note the argumen ts: DEplot( equation(s), variable(s), ind_var_range, [ init condition s in [] ], dep_var_range, options)." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "5. Here we can just sketch the dir field without giving any so lutions." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "dfieldplot(diff (p(t),t)=3*p(t)-2*p(t)^2 ,p(t), t=0..5, p=0..3);" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 101 "Answers to the questions: (a) See above. (b) 15 00 (c) 1500 (d) No. Of course you should see why." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 112 "6. Before answering any of the questions, let 's get a DEplot showing some of the stuff the questions refer to. " }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "de6:=diff(y(x),x)=x+sin(y(x ));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "DEplot(de6, y(x), x= 0..3, [[y(0)=0], [y(1)=Pi/2]], y=0..6);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "The questions: (a)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "slope:=1+sin(Pi/2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "(b) Because the slope is positive: We know " }{XPPEDIT 18 0 " -1 <= sin(y);" "6#1,$\"\"\"!\"\"-%$sinG6#%\"yG" }{TEXT -1 6 " (and " } {XPPEDIT 18 0 "sin(y) <= 1;" "6#1-%$sinG6#%\"yG\"\"\"" }{TEXT -1 17 ") so if x>1 then " }{XPPEDIT 18 0 "diff(y(x),x) = x+sin(y(x));" "6#/-%% diffG6$-%\"yG6#%\"xGF*,&F*\"\"\"-%$sinG6#-F(6#F*F," }{TEXT -1 7 " is > 0." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "(c) Just differentiate " }{XPPEDIT 18 0 "diff(y(x),x) = x+sin(y(x))" "6#/-%%diffG6$-%\"yG6#%\"x GF*,&F*\"\"\"-%$sinG6#-F(6#F*F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "diff(diff(y(x),x) = x+sin(y(x)),x);" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 15 "Now substitute." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "subs(diff(y(x),x) = x+sin(y(x)),%);" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 100 "(d) You just computed y''. Now note that y'(0)=0 and use the second derivative test. Get y''(0)=1" }{TEXT -1 0 "" }{TEXT -1 20 ", which is positive." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 186 "8 . Sort of like 6. Ask if any of the questions give you trouble after \+ some work. Here is a sketch. You have to set the stepsize smaller th an default to get a good solution curve plot." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "de8:=diff(x(t),t)=t^3-x^3;\nDEplot(de8, x(t), t =0..4, [[x(2)=1], [x(2.5)=2]], x=0..4, stepsize=0.05);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "10. (c)," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "DEplot(diff(y(x),x)=sin(x)*sin(y(x)), y(x),\nx=0..7,[ [y(0)=1], [y(0)=-1]], y=-4..4, stepsize=.05);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "17." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "de17: =diff(y(x),x)=3-y(x)+1/x;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "dfieldplot(de17, y(x), x=1..10, y=1..5); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "Sure looks like a little more than 3, but you can't re ally tell.:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "solve(3-y-1/ x, y);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "limit(%,x=infinit y);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "So 3 is porbably it." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "dsolve(de17,y(x));" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "sol:=subs(_C1=0,rhs(%));" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "limit(sol,x=infinity);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "97" 0 } {VIEWOPTS 1 1 0 1 1 1803 }