{VERSION 3 0 "SUN SPARC SOLARIS" "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 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 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 "" 11 256 1 {CSTYLE "" -1 -1 "" 0 1 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 "" 11 257 1 {CSTYLE "" -1 -1 "" 0 1 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 {EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "White test." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "1. Electrical network part. Label the m ain current " }{XPPEDIT 18 0 "I1" "6#%#I1G" }{TEXT -1 23 ", and split that into " }{XPPEDIT 18 0 "I2" "6#%#I2G" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "\011I3" "6#%#I3G" }{TEXT -1 51 " going through the ind uctor and resistor, resp. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "i1:=\{I1(t)=I2(t)+I3(t), 10=I1(t)/2+I3(t)/2+diff(I1(t),t)*2 , I3(t)/2=diff(I2(t),t)*2,\nI1(0)=0,I2(0)=0,I3(0)=0\};" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(inttrans):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 200 "### WARNING: `dsolve` has been extensively rewritten, many new result forms can occur and options are slightly d ifferent, see help page for details\nsol1:=dsolve(i1,\{I1(t),I2(t),I3( t)\},method=laplace);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Note thi s time the second argument is " }{XPPEDIT 18 0 "I1" "6#%#I1G" }{TEXT -1 73 ". [When you run the worksheet it may be different!!] The voltag e across " }{XPPEDIT 18 0 "L2" "6#%#L2G" }{TEXT -1 5 " is:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "ans1:=2*diff(rhs(sol1[2]),t) ;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "This isn't required, but it \+ is interesting anyway." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "p lot(ans1,t=0..2,y=0..10);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 120 "Cou pled mass problem. Place the origin of the two coordinate systems at \+ the natural length of all the springs. Place " }{XPPEDIT 18 0 "m1" " 6#%#m1G" }{TEXT -1 6 " at " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" } {TEXT -1 7 " and " }{XPPEDIT 18 0 "m2" "6#%#m2G" }{TEXT -1 6 " at \+ " }{XPPEDIT 18 0 "y=0" "6#/%\"yG\"\"!" }{TEXT -1 21 ". Use Newton's l aws:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "m1:=2: m2:=1: k1:= 4: k2:=2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 122 "mi1:=\{diff( x(t),t,t)*m1=-k1*x(t)+k2*(y(t)-x(t)),\ndiff(y(t),t,t)*m2=-k2*(y(t)-x(t )),\nx(0)=1, y(0)=1, D(x)(0)=0, D(y)(0)=0\};" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 194 "### WARNING: `dsolve` has been extensively rewr itten, many new result forms can occur and options are slightly differ ent, see help page for details\nmsol1:=dsolve(mi1,\{x(t),y(t)\},method =laplace);" }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 41 "Not part of the problem, but interesting:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "plot([rhs(msol1[2]),rhs(msol1[1])], t=0..10, color=[red,blue]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "He re is how much the second spring is stretched:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 42 "plot(rhs(msol1[2])-rhs(msol1[1]),t=0..10);" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "2. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "invlaplace(exp(-3*s)/(s-2),s,t);" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 39 "3. The convolution theorem says " } {XPPEDIT 18 0 "laplace(int(f(tau)*g(t-tau),tau=0..t))=F(s)*G(s)" "6#/- %(laplaceG6#-%$intG6$*&-%\"fG6#%$tauG\"\"\"-%\"gG6#,&%\"tGF/F.!\"\"F// F.;\"\"!F4*&-%\"FG6#%\"sGF/-%\"GG6#F=F/" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "invlaplace(1/s^2,s,t); invlaplace(1 /(s^2+4),s,t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "int(T*(1/ 2)*sin(2*(t-T)),T=0..t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "invlaplace(1/s^2/(s^2+4),s,t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "They turn out to be the same of course." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "4. Just let Maple do it all." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "i4:=\{diff(w(t),t,t)+9*w(t)=5*Dirac(t-Pi), w(0)=1, D(w)(0)=0\} ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 185 "### WARNING: `dsolve` has been extensively rewritten, many new result forms can occur and o ptions are slightly different, see help page for details\nsol4:=dsolve (i4,w(t),method=laplace);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "plot(rhs(sol4),t=0..5);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "5. Nothing special here." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 " i5:=\{diff(x(t),t)+diff(y(t),t)=Dirac(t), x(0)=0, diff(y(t),t)=x(t)+He aviside(t-Pi)*sin(t), y(0)=0\};" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 192 "### WARNING: `dsolve` has been extensively rewritten, many ne w result forms can occur and options are slightly different, see help \+ page for details\nsol5:=dsolve(i5,\{x(t),y(t)\},method=laplace);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "plot([rhs(sol5[2]), rhs(sol5 [1])], t=0..2*Pi, color=[red,blue]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "Kinda cute..." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 12 "Yellow test." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "1. Electrical network part. Label the main current " }{XPPEDIT 18 0 " I1" "6#%#I1G" }{TEXT -1 23 ", and split that into " }{XPPEDIT 18 0 "I 2" "6#%#I2G" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "\011I3" "6#%#I3G" } {TEXT -1 51 " going through the inductor and resistor, resp. " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "i1:=\{I1(t)=I2(t)+I3(t), \+ 10=I1(t)*2+I3(t)*2+diff(I1(t),t)/2, I3(t)*2=diff(I2(t),t)/2,\nI1 (0)=0,I2(0)=0,I3(0)=0\};" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(inttrans):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 200 "### WA RNING: `dsolve` has been extensively rewritten, many new result forms \+ can occur and options are slightly different, see help page for detail s\nsol1:=dsolve(i1,\{I1(t),I2(t),I3(t)\},method=laplace);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "Note this time the first argument is " } {XPPEDIT 18 0 "I1" "6#%#I1G" }{TEXT -1 74 ". [When you run the worksh eet it may be different!!] The voltage across " }{XPPEDIT 18 0 "L2" " 6#%#L2G" }{TEXT -1 5 " is:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "ans1:=(1/2)*diff(rhs(sol1[2]),t);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "This isn't required, but it is interesting anyway." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "plot(ans1,t=0..2);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 120 "Coupled mass problem. Place the \+ origin of the two coordinate systems at the natural length of all the \+ springs. Place " }{XPPEDIT 18 0 "m1" "6#%#m1G" }{TEXT -1 6 " at " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "m2" "6#%#m2G" }{TEXT -1 6 " at " }{XPPEDIT 18 0 "y=0" "6#/%\"y G\"\"!" }{TEXT -1 21 ". Use Newton's laws:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "m1:=1: m2:=3: k1:=2: k2:=4:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 122 "mi1:=\{diff(x(t),t,t)*m1=-k1*x(t)+k2*(y(t)- x(t)),\ndiff(y(t),t,t)*m2=-k2*(y(t)-x(t)),\nx(0)=1, y(0)=1, D(x)(0)=0, D(y)(0)=0\};" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 187 "### WARNI NG: `dsolve` has been extensively rewritten, many new result forms can occur and options are slightly different, see help page for details\n dsolve(mi1,\{x(t),y(t)\},method=laplace);" }}{PARA 257 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "msol1:=evalf(%); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "Not part of the problem, but \+ interesting:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "plot([rhs(m sol1[2]),rhs(msol1[1])],t=0..10, color=[red,blue]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "Here is how much the second spring is stretched :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "plot(rhs(msol1[2])-rhs (msol1[1]),t=0..10);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "2. Use p iecewise and then convert:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "piecewise(t<1,0,1 " 0 "" {MPLTEXT 1 0 21 "convert(%,Heaviside);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "laplace(%,t,s);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "3. Maple will use the convolution theorem to represent the answer if you use laplace." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "i3: =\{diff(y(t),t,t)+4*y(t)=g(t), y(0)=1, D(y)(0)=0\};" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 179 "### WARNING: `dsolve` has been extensive ly rewritten, many new result forms can occur and options are slightly different, see help page for details\ndsolve(i3,y(t),method=laplace); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "4. Just let Maple do it all. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "is4:=\{diff(x(t),t)-di ff(y(t),t)=Heaviside(t-Pi)*sin(t), x(0)=0, diff(y(t),t)=x(t)+Dirac(t), y(0)=0\};" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 191 "### WARNING: `dsolve` has been extensively rewritten, many new result forms can oc cur and options are slightly different, see help page for details\ns4: =dsolve(is4,\{x(t),y(t)\},method=laplace);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 54 "plot([rhs(s4[1]),rhs(s4[2])],t=0..6,color=[red,blue ]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 129 "Note the blue plot lies o n top of the red plot. [These plots are not necessary, but are fun.] \+ Here is a plot of the difference:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "plot(rhs(s4[1])-rhs(s4[2]),t=0..16);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "5. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "i5:=\{diff(w(t),t,t)+w(t)=Dirac(t-Pi), w(0)=0, D(w)(0 )=1\};" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 185 "### WARNING: `ds olve` has been extensively rewritten, many new result forms can occur \+ and options are slightly different, see help page for details\nsol5:=d solve(i5,w(t),method=laplace);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "plot(rhs(sol5),t=0..10);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 219 "My total tim e to do both tests, including the unnecessary plots, was 22 minutes. \+ This, of course, did not include whatever time it takes to copy the st uff from the screen to your paper. I know that can be substantial." } }}}{MARK "70" 0 }{VIEWOPTS 1 1 0 1 1 1803 }