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}1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 259 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 260 1 {CSTYLE "" -1 -1 "T imes" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Normal" -1 261 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 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 262 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 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 26 " Math 217 Fall 2000 Exam \+ 2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 277 46 "Notational Remark: In this exam, the sy mbol " }{XPPEDIT 281 1 "diff(y(x),x)" "6#-%%diffG6$-%\"yG6#%\"xGF)" } {TEXT 279 9 " means " }{XPPEDIT 282 0 "dy/dx;" "6#*&%#dyG\"\"\"%#dxG !\"\"" }{TEXT 278 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 355 96 "1. If f(t) = 1 for all t > 0 and if F is the Laplace transform of f, then what is F(10)?" }}{PARA 0 " " 0 "" {TEXT 356 0 "" }}{PARA 257 "" 0 "" {TEXT -1 97 "a) 1 \+ b) 10 c) 5 d) 0.1 e) 0.2 \+ " }}{PARA 257 "" 0 "" {TEXT -1 18 "f) 1/10! g) " } {XPPEDIT 18 0 "1/GAMMA(10);" "6#*&\"\"\"F$-%&GAMMAG6#\"#5!\"\"" } {TEXT -1 8 " h) " }{XPPEDIT 19 1 "GAMMA(10);" "6#-%&GAMMAG6#\"#5" }{TEXT -1 28 " i) 1/101 j) 1/100" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 685 3 "(d)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(inttrans):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "laplace(1,t,s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*& \"\"\"F$%\"sG!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "subs( s=10,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"\"\"#5" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 357 41 "2. What is th e least positive eigenvalue " }{XPPEDIT 258 0 "lambda;" "6#%'lambdaG" }{TEXT 358 25 " of the endpoint problem " }{XPPEDIT 359 1 "y*`''`(x)+l ambda*y(x) = 0,y(Pi) = 0,y(-Pi) = 0;" "6%/,&*&%\"yG\"\"\"-%#''G6#%\"xG F'F'*&%'lambdaGF'-F&6#F+F'F'\"\"!/-F&6#%#PiGF0/-F&6#,$F4!\"\"F0" } {TEXT 360 2 " ?" }}{PARA 3 "" 0 "" {TEXT 283 64 "a) 1/16 b) 1/8 c) 1/4 d) 1/2 e) 1/" }{XPPEDIT 361 0 "sqrt(2); " "6#-%%sqrtG6#\"\"#" }}{PARA 3 "" 0 "" {TEXT 284 3 "f) " }{XPPEDIT 362 0 "1/Pi;" "6#*&\"\"\"F$%#PiG!\"\"" }{TEXT 287 14 " g) " }{XPPEDIT 363 0 "1/sqrt(Pi);" "6#*&\"\"\"F$-%%sqrtG6#%#PiG!\"\"" } {TEXT 286 7 " h) " }{XPPEDIT 364 0 "2/sqrt(Pi);" "6#*&\"\"#\"\"\"-% %sqrtG6#%#PiG!\"\"" }{TEXT 288 9 " i) " }{XPPEDIT 365 0 "Pi/2;" " 6#*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT 289 13 " j) " }{XPPEDIT 366 0 "Pi/4;" "6#*&%#PiG\"\"\"\"\"%!\"\"" }{TEXT 285 1 " " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 367 28 "3. The graph of a function " }{XPPEDIT 368 0 "proc (t) options operator, arrow; x(t) end proc;" "6#R6#%\"tG7\"6$%)o peratorG%&arrowG6\"-%\"xG6#F%F*F*F*" }{TEXT 290 1 " " }{TEXT 369 31 " \+ is plotted below. What might " }{XPPEDIT 370 0 "f(t);" "6#-%\"fG6#%\" tG" }{TEXT 371 10 " be if " }{XPPEDIT 372 1 "x(t);" "6#-%\"xG6#%\"t G" }{TEXT 373 39 " satisfies the differential equation " }{XPPEDIT 374 1 "diff(x(t),`$`(t,2))+25*x(t) = f(t);" "6#/,&-%%diffG6$-%\"xG6#% \"tG-%\"$G6$F+\"\"#\"\"\"*&\"#DF0-F)6#F+F0F0-%\"fG6#F+" }{TEXT 375 10 " and if " }{XPPEDIT 376 1 "x(t);" "6#-%\"xG6#%\"tG" }{TEXT 377 24 " has the following graph" }}{PARA 13 "" 1 "" {GLPLOT2D 246 185 185 {PLOTDATA 2 "6'-%'CURVESG6$7i[l7$$\"\"!F)F(7$$\"3emmm;arz@!#=$\"3!*[du plk[AF-7$$\"39LLLL3VfVF-$\"3]^$[vcP.#RF-7$$\"3t**\\i:0dL[F-$\"3PK,&*)[ .)3TF-7$$\"3Kmm\"z>5xI&F-$\"3:J]BX`(p@%F-7$$\"3!HL3-))\\=y&F-$\"3\"*>o %e'y[RUF-7$$\"3]****\\i&*)fD'F-$\"3e-Y\\@drsTF-7$$\"3mKL3F*oU?(F-$\"3# zI*zK+ImPF-7$$\"3%pmm;H[D:)F-$\"3:^B0)\\Ix+$F-7$$\"3'****\\PpU&G5!#<$ \"3-t>!QDxL#G!#>7$$\"3MLLLe0$=C\"FV$!3?VORB,gyHF-7$$\"3SLL$eR\"=\\8FV$ !3AS([zjzIR%F-7$$\"3[LLLLA`c9FV$!3iXx:1vrDaF-7$$\"3TLL3_w?5:FV$!3cJf@; 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"6#-%$cosG6#*&\"\"&\"\"\"%\"tGF(" }{TEXT 384 7 " \+ d) " }{XPPEDIT 385 1 "sin(25*t);" "6#-%$sinG6#*&\"#D\"\"\"%\"tGF(" } {TEXT 386 7 " e) " }{XPPEDIT 387 1 "exp(5*t);" "6#-%$expG6#*&\"\"& \"\"\"%\"tGF(" }{TEXT 388 5 "\n\nf) " }{XPPEDIT 389 1 "exp(-5*t);" "6# -%$expG6#,$*&\"\"&\"\"\"%\"tGF)!\"\"" }{TEXT 390 6 " g) " }{XPPEDIT 391 1 "exp(sqrt(5)*t);" "6#-%$expG6#*&-%%sqrtG6#\"\"&\"\"\"%\"tGF+" } {TEXT 392 8 " h) " }{XPPEDIT 393 1 "exp(-sqrt(5)*t);" "6#-%$expG6# ,$*&-%%sqrtG6#\"\"&\"\"\"%\"tGF,!\"\"" }{TEXT 394 5 " i) " }{XPPEDIT 395 1 "sin(sqrt(5));" "6#-%$sinG6#-%%sqrtG6#\"\"&" }{TEXT 396 9 " \+ j) " }{XPPEDIT 397 1 "cos(sqrt(5)*t);" "6#-%$cosG6#*&-%%sqrtG6#\"\"& \"\"\"%\"tGF+" }{TEXT 398 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 280 0 "" }{TEXT 694 5 "(c) \n" }}{PARA 0 "" 0 "" {TEXT -1 47 "Natural frequency equals the forcing frequency. " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 399 8 "4. Let " }{XPPEDIT 400 1 "y (x) = Sum(c[n]*(x-4)^n,n = 0 .. infinity);" "6#/-%\"yG6#%\"xG-%$SumG6$ *&&%\"cG6#%\"nG\"\"\"),&F'F0\"\"%!\"\"F/F0/F/;\"\"!%)infinityG" } {TEXT 401 53 " denote a power series expansion of a solution of " } {XPPEDIT 402 1 "(x^2+9)*y*`''`(x)+y(x)+x^3*y(x) = 0;" "6#/,(*(,&*$%\"x G\"\"#\"\"\"\"\"*F*F*%\"yGF*-%#''G6#F(F*F*-F,6#F(F**&F(\"\"$-F,6#F(F*F *\"\"!" }{TEXT 403 79 ". Then what is the largest radius of convergen ce of this power series that is " }{TEXT 681 10 "guaranteed" }{TEXT 682 112 " by the theorem on solutions at ordinary points? \n\na) 0 \+ b) 1 c) 2 d) 3 e) 4 " }}{PARA 3 "" 0 "" {TEXT 291 55 "f) 5 g) 6 h) 9 i) 16 j) \+ 25" }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 696 3 "(f) " }}{PARA 0 "" 0 "" {TEXT -1 69 "There is a singular point at 0 + 3*i. The distance to 4 + 0*i is 5." }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 404 27 "5. What is the coefficient " }{XPPEDIT 405 1 "c[4];" "6#&%\"cG 6#\"\"%" }{TEXT 406 35 " in the Maclaurin series expansion " } {XPPEDIT 407 1 "cos(x)^2 = Sum(c[n]*x^n,n = 0 .. infinity);" "6#/*$-%$ cosG6#%\"xG\"\"#-%$SumG6$*&&%\"cG6#%\"nG\"\"\")F(F1F2/F1;\"\"!%)infini tyG" }{TEXT 408 5 ". " }}{PARA 3 "" 0 "" {TEXT 422 34 " (This is th e square of cosine of " }{XPPEDIT 409 1 "x;" "6#%\"xG" }{TEXT 410 20 " , not the cosine of " }{XPPEDIT 411 1 "x^2;" "6#*$%\"xG\"\"#" }{TEXT 412 10 ". ) " }}{PARA 0 "" 0 "" {TEXT 413 0 "" }}{PARA 0 "" 0 " " {TEXT 414 0 "" }}{PARA 258 "" 0 "" {TEXT -1 4 "a) " }{XPPEDIT 19 1 "1/2;" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT -1 13 " b) " } {XPPEDIT 19 1 "-1/2;" "6#,$*&\"\"\"F%\"\"#!\"\"F'" }{TEXT -1 15 " \+ c) " }{XPPEDIT 19 1 "1/3;" "6#*&\"\"\"F$\"\"$!\"\"" }{TEXT -1 12 " d) " }{XPPEDIT 19 1 "-1/3;" "6#,$*&\"\"\"F%\"\"$!\"\"F'" }{TEXT -1 13 " e) " }{XPPEDIT 19 1 "2/3;" "6#*&\"\"#\"\"\"\" \"$!\"\"" }{TEXT -1 5 "\n\nf) " }{XPPEDIT 19 1 "-2/3;" "6#,$*&\"\"#\" \"\"\"\"$!\"\"F(" }{TEXT -1 15 " g) " }{XPPEDIT 19 1 "1/6; " "6#*&\"\"\"F$\"\"'!\"\"" }{TEXT -1 14 " h) " }{XPPEDIT 19 1 "-1/6;" "6#,$*&\"\"\"F%\"\"'!\"\"F'" }{TEXT -1 13 " i) " } {XPPEDIT 19 1 "5/24;" "6#*&\"\"&\"\"\"\"#C!\"\"" }{TEXT -1 13 " \+ j) " }{XPPEDIT 19 1 "-5/24;" "6#,$*&\"\"&\"\"\"\"#C!\"\"F(" }{TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 695 3 "(c)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "series(cos(x),x=0,5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#++%\"xG\"\"\"\"\"!#!\"\"\"\"#F)#F%\" #C\"\"%-%\"OG6#F%\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "c onvert(%,polynom);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(\"\"\"F$*&#F$ \"\"#F$*$)%\"xGF'F$F$!\"\"*&#F$\"#CF$)F*\"\"%F$F$" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 14 "expand( %^2 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,\"\"\"F$*$)%\"xG\"\"#F$!\"\"*&#F$\"\"$F$)F'\"\"%F$F$*&#F$\"#CF $*$)F'\"\"'F$F$F)*&#F$\"$w&F$)F'\"\")F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 415 9 " 6. Let " }{XPPEDIT 416 1 "y(x) = Sum(c[n]*x^n,n = 0 .. infinity);" "6#/-%\"yG6#%\"xG-%$Su mG6$*&&%\"cG6#%\"nG\"\"\")F'F/F0/F/;\"\"!%)infinityG" }{TEXT 417 78 " \+ denote the power series of the unique solution of the initial value pr oblem " }{XPPEDIT 418 1 "y*`'`(x) = y(x)+x^3,y(0) = 4;" "6$/*&%\"yG\" \"\"-%\"'G6#%\"xGF&,&-F%6#F*F&*$F*\"\"$F&/-F%6#\"\"!\"\"%" }{TEXT 419 25 " . What is the value of " }{XPPEDIT 420 1 "c[2];" "6#&%\"cG6#\"\" #" }{TEXT 421 106 "? \n\na) 1 b) -1 c) 1/2 d) -1/2 \+ e) 2 f) -2 g) 3 h) -3 i) 4 j) -4" }}{PARA 3 " " 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 4 "(e)\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "dsolve(\{diff(y(x),x) = y(x)+x^3,y( 0)=4\},y(x),series);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG +1F'\"\"%\"\"!F)\"\"\"\"\"#F,#F,\"\"$F.#\"\"&\"#7F)#F+F1F0-%\"OG6#F+\" \"'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 423 9 "7. Let " }{XPPEDIT 424 1 "y(x) = Sum(c[ n]*x^n,n = 0 .. infinity);" "6#/-%\"yG6#%\"xG-%$SumG6$*&&%\"cG6#%\"nG \"\"\")F'F/F0/F/;\"\"!%)infinityG" }{TEXT 425 69 " denote the power se ries of a solution of the differential equation " }{XPPEDIT 426 1 "y* `''`(x)+x*y*`'`(x)+2*y(x) = 0;" "6#/,(*&%\"yG\"\"\"-%#''G6#%\"xGF'F'*( F+F'F&F'-%\"'G6#F+F'F'*&\"\"#F'-F&6#F+F'F'\"\"!" }{TEXT 427 76 " . Th en the coefficients satisfy which of the following recurrences?\n\na) \+ " }{XPPEDIT 257 0 "c[n+1] = c[n]/(n+1);" "6#/&%\"cG6#,&%\"nG\"\"\"F) F)*&&F%6#F(F),&F(F)F)F)!\"\"" }{TEXT 292 26 " b) \+ " }{XPPEDIT 259 0 "c[n+1] = (2+c[n])/(n+1);" "6#/&%\"cG6#,&%\"nG\"\" \"F)F)*&,&\"\"#F)&F%6#F(F)F),&F(F)F)F)!\"\"" }{TEXT 293 8 "\n\nc) \+ " }{XPPEDIT 261 0 "c[n+1] = -c[n]/(n+1);" "6#/&%\"cG6#,&%\"nG\"\"\"F)F ),$*&&F%6#F(F),&F(F)F)F)!\"\"F/" }{TEXT 294 21 " d) \+ " }{XPPEDIT 263 0 "c[n+1] = -(2+c[n])/(n+1);" "6#/&%\"cG6#,&%\"nG\"\" \"F)F),$*&,&\"\"#F)&F%6#F(F)F),&F(F)F)F)!\"\"F1" }{TEXT 295 7 "\n\ne) \+ " }{XPPEDIT 265 0 "c[n+1] = c[n]/(n*(n+1));" "6#/&%\"cG6#,&%\"nG\"\" \"F)F)*&&F%6#F(F)*&F(F),&F(F)F)F)F)!\"\"" }{TEXT 296 19 " \+ f) " }{XPPEDIT 267 0 "c[n+2] = c[n]/(n+1);" "6#/&%\"cG6#,&%\"nG\"\" \"\"\"#F)*&&F%6#F(F),&F(F)F)F)!\"\"" }{TEXT 297 6 "\n\ng) " } {XPPEDIT 269 0 "c[n+2] = -c[n]/(n+1);" "6#/&%\"cG6#,&%\"nG\"\"\"\"\"#F ),$*&&F%6#F(F),&F(F)F)F)!\"\"F0" }{TEXT 298 23 " h) \+ " }{XPPEDIT 271 0 "c[n+2] = c[n]/(n+2);" "6#/&%\"cG6#,&%\"nG\"\"\"\" \"#F)*&&F%6#F(F),&F(F)F*F)!\"\"" }{TEXT 299 7 "\n\ni) " }{XPPEDIT 273 0 "c[n+2] = -c[n]/(n+2);" "6#/&%\"cG6#,&%\"nG\"\"\"\"\"#F),$*&&F%6 #F(F),&F(F)F*F)!\"\"F0" }{TEXT 300 22 " j) " } {XPPEDIT 275 0 "c[n+2] = c[n]/((n+1)*(n+2));" "6#/&%\"cG6#,&%\"nG\"\" \"\"\"#F)*&&F%6#F(F)*&,&F(F)F)F)F),&F(F)F*F)F)!\"\"" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 693 3 "(c)" }{TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(Slode);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7-%,D EdetermineG%)FPseriesG%)FTseriesG%2candidate_mpointsG%1candidate_point sG%5hypergeom_formal_solG%5hypergeom_series_solG%6mhypergeom_series_so lG%3msparse_series_solG%6polynomial_series_solG%4rational_series_solG " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "FPseries( diff(y(x),x$2 ) + x*diff(y(x),x)+2*y(x) = 0, y(x), c(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%*FPSstructG6$,(&%#_CG6#\"\"!\"\"\"*&&F(6#F+F+%\"xGF+F +-%$SumG6$*&-%\"cG6#%\"nGF+)F/F7F+/F7;\"\"#%)infinityGF+,&*&,&*$)F7F;F +F+F7!\"\"F+F4F+F+*&F7F+-F56#,&F7F+F;FBF+F+-%&TABLEG6#7)/%\"MGF+/%$rec G-%&RESolG6&<#/,&*&,&F7F+F;F+F+F4F+F+*&,(F@F+*&\"\"$F+F7F+F+F;F+F+-F56 #FVF+F+F*<#F4<$/-F5F)Fjn/-F5F.F\\o%%INFOG/%&pointGF*/%$P_nGF8/%%mvarGF //%(initialG-FH6#7$/F*F'/F+F-/%-arb_constantGF(" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 43 "c(n) = solve(n*c(n-2)+(n^2-n)*c(n)=0,c(n));" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"cG6#%\"nG,$*&-F%6#,&F'\"\"\"\"\" #!\"\"F-,&F'F-F-F/F/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "s ubs(n=n+2,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"cG6#,&%\"nG\"\" \"\"\"#F),$*&-F%6#F(F),&F(F)F)F)!\"\"F0" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 428 100 "8. For which of the following differential eq uations is 0 a regular singular point?\n\nI) " }{XPPEDIT 429 0 "x*y*`''`(x)+cos(x)*y*`'`(x)+x*y(x) = 0;" "6#/,(*(%\"xG\"\"\"%\" yGF'-%#''G6#F&F'F'*(-%$cosG6#F&F'F(F'-%\"'G6#F&F'F'*&F&F'-F(6#F&F'F'\" \"!" }{TEXT 430 16 "\nII) " }{XPPEDIT 431 0 "y*`''`(x)+sin( x)*y*`'`(x)/x+y(x) = 0;" "6#/,(*&%\"yG\"\"\"-%#''G6#%\"xGF'F'**-%$sinG 6#F+F'F&F'-%\"'G6#F+F'F+!\"\"F'-F&6#F+F'\"\"!" }}{PARA 3 "" 0 "" {TEXT 432 15 "III) " }{XPPEDIT 433 0 "x^2*y*`''`(x)+2*sin(x) *y*`'`(x)+y(x) = 0;" "6#/,(*(%\"xG\"\"#%\"yG\"\"\"-%#''G6#F&F)F)**F'F) -%$sinG6#F&F)F(F)-%\"'G6#F&F)F)-F(6#F&F)\"\"!" }{TEXT 434 1 "\n" }} {PARA 3 "" 0 "" {TEXT 301 195 "a) I only b) II only \+ c) III only d) I, II only e) I, III only \n f) II, III only g) I, II, and III h) None of them (No choi ces (i) and (j).)" }}{PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 262 "" 0 "" {TEXT -1 4 "(e)\n" }}{PARA 0 "" 0 "" {TEXT -1 79 "0 is a regular singular point of (I) and (III) but an ordinary point of (II). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 435 154 "9. When we seek a Frobenius series centered at 0 for a solution, which of the following equations is the indicial equation of the differential equa tion " }{XPPEDIT 436 0 "3*x*y*`''`(x)+y*`'`(x)-y(x) = 0;" "6#/,(**\" \"$\"\"\"%\"xGF'%\"yGF'-%#''G6#F(F'F'*&F)F'-%\"'G6#F(F'F'-F)6#F(!\"\" \"\"!" }{TEXT 437 6 "?\n\na) " }{XPPEDIT 257 1 "r^2-r/2 = 0;" "6#/,&*$ %\"rG\"\"#\"\"\"*&F&F(F'!\"\"F*\"\"!" }{TEXT 302 10 " b) " } {XPPEDIT 259 1 "r^2-r/2-1 = 0;" "6#/,(*$%\"rG\"\"#\"\"\"*&F&F(F'!\"\"F *F(F*\"\"!" }{TEXT 303 10 " c) " }{XPPEDIT 261 1 "r^2-2*r/3 = 0; " "6#/,&*$%\"rG\"\"#\"\"\"*(F'F(F&F(\"\"$!\"\"F+\"\"!" }{TEXT 304 10 " d) " }{XPPEDIT 263 1 "r^2+r/2-1 = 0;" "6#/,(*$%\"rG\"\"#\"\"\"* &F&F(F'!\"\"F(F(F*\"\"!" }{TEXT 305 10 " e) " }{XPPEDIT 265 1 "r /3-1 = 0;" "6#/,&*&%\"rG\"\"\"\"\"$!\"\"F'F'F)\"\"!" }{TEXT 306 6 "\n \nf) " }{XPPEDIT 267 1 "x*r^2+r/3-1 = 0;" "6#/,(*&%\"xG\"\"\"*$%\"rG \"\"#F'F'*&F)F'\"\"$!\"\"F'F'F-\"\"!" }{TEXT 307 13 " g) " } {XPPEDIT 274 0 "r^2-3*r/2+1 = 0;" "6#/,(*$%\"rG\"\"#\"\"\"*(\"\"$F(F&F (F'!\"\"F+F(F(\"\"!" }{TEXT 310 12 " h) " }{XPPEDIT 269 1 "r^2 +r/2-x*r = 0;" "6#/,(*$%\"rG\"\"#\"\"\"*&F&F(F'!\"\"F(*&%\"xGF(F&F(F* \"\"!" }{TEXT 308 12 " i) " }{XPPEDIT 271 1 "-r/3-1 = 0;" "6#/ ,&*&%\"rG\"\"\"\"\"$!\"\"F)F'F)\"\"!" }{TEXT 309 94 "\n\nj) There is n o indicial equation because 0 is not a regular singular point of the e quation.\n" }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 698 3 "(c)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 45 "3*x*diff(y(x),x$2) + diff(y(x),x) - y(x) = 0;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&%\"xG\"\"\"-%%diffG6$-%\"yG6#F&-% \"$G6$F&\"\"#F'\"\"$-F)6$F+F&F'F+!\"\"\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "%*x/3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*&%\"x G\"\"\",(*&F&F'-%%diffG6$-%\"yG6#F&-%\"$G6$F&\"\"#F'\"\"$-F+6$F-F&F'F- !\"\"F'#F'F4\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "expand (%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&)%\"xG\"\"#\"\"\"-%%diffG 6$-%\"yG6#F'-%\"$G6$F'F(F)F)*(#F)\"\"$F)F'F)-F+6$F-F'F)F)*&#F)F5F)*&F' F)F-F)F)!\"\"\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "indic ial_eqn := r*(r-1) + 1/3*r + (-1/3*0) = 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-indicial_eqnG/,&*&%\"rG\"\"\",&F(F)F)!\"\"F)F)*&#F) \"\"$F)F(F)F)\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "expan d(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*$)%\"rG\"\"#\"\"\"F)*&#F( \"\"$F)F'F)!\"\"\"\"!" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " } {TEXT 438 49 "10. Which one of the following Frobenius series " } {XPPEDIT 256 0 "y(x) = Sum(c[n]*x^(n+r),n = 0 .. infinity),c[0] <> 0; " "6$/-%\"yG6#%\"xG-%$SumG6$*&&%\"cG6#%\"nG\"\"\")F',&F/F0%\"rGF0F0/F/ ;\"\"!%)infinityG0&F-6#F6F6" }{TEXT 680 53 " might be a solution of t he differential equation " }{XPPEDIT 439 1 "d^2*y/(d*x^2)-2*dy/(x*dx )+9*y/(4*x^2) = 0;" "6#/,(*(%\"dG\"\"#%\"yG\"\"\"*&F&F)*$%\"xGF'F)!\" \"F)*(F'F)%#dyGF)*&F,F)%#dxGF)F-F-*(\"\"*F)F(F)*&\"\"%F)*$F,F'F)F-F)\" \"!" }{TEXT 440 10 " ?\n\na) " }{XPPEDIT 258 1 "y(x) = Sum(c[n]*x^( n-2),n = 0 .. infinity);" "6#/-%\"yG6#%\"xG-%$SumG6$*&&%\"cG6#%\"nG\" \"\")F',&F/F0\"\"#!\"\"F0/F/;\"\"!%)infinityG" }{TEXT 311 12 " \+ b) " }{XPPEDIT 276 0 "y(x) = Sum(c[n]*x^(n-3/2),n = 0 .. infinity);" "6#/-%\"yG6#%\"xG-%$SumG6$*&&%\"cG6#%\"nG\"\"\")F',&F/F0*&\"\"$F0\"\"# !\"\"F6F0/F/;\"\"!%)infinityG" }{TEXT 319 1 " " }{TEXT 320 4 " " }} {PARA 3 "" 0 "" {TEXT 441 3 "c) " }{XPPEDIT 260 1 "y(x) = Sum(c[n]*x^( n-1),n = 0 .. infinity);" "6#/-%\"yG6#%\"xG-%$SumG6$*&&%\"cG6#%\"nG\" \"\")F',&F/F0F0!\"\"F0/F/;\"\"!%)infinityG" }{TEXT 312 14 " \+ d) " }{XPPEDIT 262 1 "y(x) = Sum(c[n]*x^(n-1/2),n = 0 .. infinity);" " 6#/-%\"yG6#%\"xG-%$SumG6$*&&%\"cG6#%\"nG\"\"\")F',&F/F0*&F0F0\"\"#!\" \"F5F0/F/;\"\"!%)infinityG" }{TEXT 313 7 " " }}{PARA 3 "" 0 "" {TEXT 442 3 "e) " }{XPPEDIT 264 1 "y(x) = Sum(c[n]*x^(n+1/4),n = 0 .. \+ infinity);" "6#/-%\"yG6#%\"xG-%$SumG6$*&&%\"cG6#%\"nG\"\"\")F',&F/F0*& F0F0\"\"%!\"\"F0F0/F/;\"\"!%)infinityG" }{TEXT 314 11 " f) " } {XPPEDIT 266 1 "y(x) = Sum(c[n]*x^(n+1/2),n = 0 .. infinity);" "6#/-% \"yG6#%\"xG-%$SumG6$*&&%\"cG6#%\"nG\"\"\")F',&F/F0*&F0F0\"\"#!\"\"F0F0 /F/;\"\"!%)infinityG" }{TEXT 315 5 "\n\ng) " }{XPPEDIT 268 1 "y(x) = S um(c[n]*x^(n+3/4),n = 0 .. infinity);" "6#/-%\"yG6#%\"xG-%$SumG6$*&&% \"cG6#%\"nG\"\"\")F',&F/F0*&\"\"$F0\"\"%!\"\"F0F0/F/;\"\"!%)infinityG " }{TEXT 316 11 " h) " }{XPPEDIT 270 1 "y(x) = Sum(c[n]*x^(n+5/ 4),n = 0 .. infinity);" "6#/-%\"yG6#%\"xG-%$SumG6$*&&%\"cG6#%\"nG\"\" \")F',&F/F0*&\"\"&F0\"\"%!\"\"F0F0/F/;\"\"!%)infinityG" }{TEXT 317 5 " " }}{PARA 3 "" 0 "" {TEXT 443 4 "i) " }{XPPEDIT 272 1 "y(x) = Su m(c[n]*x^(n+3/2),n = 0 .. infinity);" "6#/-%\"yG6#%\"xG-%$SumG6$*&&%\" cG6#%\"nG\"\"\")F',&F/F0*&\"\"$F0\"\"#!\"\"F0F0/F/;\"\"!%)infinityG" } {TEXT 318 11 " j) " }{XPPEDIT 256 0 "y(x) = Sum(c[n]*x^(n+5/2), n = 0 .. infinity);" "6#/-%\"yG6#%\"xG-%$SumG6$*&&%\"cG6#%\"nG\"\"\")F ',&F/F0*&\"\"&F0\"\"#!\"\"F0F0/F/;\"\"!%)infinityG" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 700 3 "(i)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "diff(y(x),x$2) - (2/x)*diff(y(x),x) + (9/(4*x^2))*y(x ) = 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(-%%diffG6$-%\"yG6#%\"xG-% \"$G6$F+\"\"#\"\"\"*&*&F/F0-F&6$F(F+F0F0F+!\"\"F5*&*&#\"\"*\"\"%F0F(F0 F0*$)F+F/F0F5F0\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "x^2* %;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%\"xG\"\"#\"\"\",(-%%diffG6$ -%\"yG6#F&-%\"$G6$F&F'F(*&*&F'F(-F+6$F-F&F(F(F&!\"\"F7*&*&#\"\"*\"\"%F (F-F(F(*$F%F(F7F(F(\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "expand(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&)%\"xG\"\"#\"\"\"- %%diffG6$-%\"yG6#F'-%\"$G6$F'F(F)F)*(F(F)F'F)-F+6$F-F'F)!\"\"*&#\"\"* \"\"%F)F-F)F)\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "indic ial_eqn := r*(r-1)-2*r+9/4 = 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%- indicial_eqnG/,(*&%\"rG\"\"\",&F(F)F)!\"\"F)F)*&\"\"#F)F(F)F+#\"\"*\" \"%F)\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "solve(indicia l_eqn);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$#\"\"$\"\"#F#" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 444 8 "11. If \+ " }{XPPEDIT 445 1 "F(s) = (2*s+18)/(s^2+9);" "6#/-%\"FG6#%\"sG*&,&*&\" \"#\"\"\"F'F,F,\"#=F,F,,&*$F'F+F,\"\"*F,!\"\"" }{TEXT 446 10 " and if " }{XPPEDIT 447 1 "F;" "6#%\"FG" }{TEXT 448 31 " is the Laplace tra nsform of " }{XPPEDIT 449 1 "f;" "6#%\"fG" }{TEXT 450 16 ", then what is " }{XPPEDIT 451 1 "f(Pi/18);" "6#-%\"fG6#*&%#PiG\"\"\"\"#=!\"\"" }{TEXT 452 2 " ?" }}{PARA 3 "" 0 "" {TEXT 453 4 "a) " }{XPPEDIT 454 1 "sqrt(3)+3/2;" "6#,&-%%sqrtG6#\"\"$\"\"\"*&F'F(\"\"#!\"\"F(" }{TEXT 455 7 " b) " }{XPPEDIT 456 1 "2*sqrt(3)+3/2;" "6#,&*&\"\"#\"\"\"-%% sqrtG6#\"\"$F&F&*&F*F&F%!\"\"F&" }{TEXT 457 7 " c) " }{XPPEDIT 458 1 "2*sqrt(3)+3;" "6#,&*&\"\"#\"\"\"-%%sqrtG6#\"\"$F&F&F*F&" }{TEXT 459 9 " d) " }{XPPEDIT 460 1 "3*sqrt(3)+3;" "6#,&*&\"\"$\"\"\"-%% sqrtG6#F%F&F&F%F&" }{TEXT 461 7 " e) " }{XPPEDIT 462 1 "3*sqrt(3)+2 ;" "6#,&*&\"\"$\"\"\"-%%sqrtG6#F%F&F&\"\"#F&" }{TEXT 463 2 " " }} {PARA 3 "" 0 "" {TEXT 464 3 "f) " }{XPPEDIT 465 1 "sqrt(3)/2+3;" "6#,& *&-%%sqrtG6#\"\"$\"\"\"\"\"#!\"\"F)F(F)" }{TEXT 466 7 " g) " } {XPPEDIT 467 1 "sqrt(2)+3;" "6#,&-%%sqrtG6#\"\"#\"\"\"\"\"$F(" }{TEXT 468 7 " h) " }{XPPEDIT 469 1 "sqrt(2)+2;" "6#,&-%%sqrtG6#\"\"#\"\" \"F'F(" }{TEXT 470 7 " i) " }{XPPEDIT 471 1 "sqrt(3)+2;" "6#,&-%%sq rtG6#\"\"$\"\"\"\"\"#F(" }{TEXT 472 9 " j) " }{XPPEDIT 473 1 "sqr t(3)+3;" "6#,&-%%sqrtG6#\"\"$\"\"\"F'F(" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 4 "(j) \n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(inttrans);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#7/%)addtableG%(fourierG%+fouriercosG%+ fouriersinG%'hankelG%(hilbertG%+invfourierG%+invhilbertG%+invlaplaceG% *invmellinG%(laplaceG%'mellinG%*savetableG" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 33 "invlaplace((2*s+18)/(s^2+9),s,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$cosG6#,$%\"tG\"\"$\"\"#*&\"\"'\"\"\"-%$sinGF&F- F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "subs(t=Pi/18,%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$cosG6#,$%#PiG#\"\"\"\"\"'\"\"#*&F +F*-%$sinGF&F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplif y(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$-%%sqrtG6#\"\"$\"\"\"F)F( F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 321 8 "12. If " }{XPPEDIT 322 1 "F(s) = (2*s+9)/(s^2+ 4*s+5);" "6#/-%\"FG6#%\"sG*&,&*&\"\"#\"\"\"F'F,F,\"\"*F,F,,(*$F'F+F,*& \"\"%F,F'F,F,\"\"&F,!\"\"" }{TEXT 323 10 " and if " }{XPPEDIT 324 1 "F;" "6#%\"FG" }{TEXT 325 31 " is the Laplace transform of " } {XPPEDIT 326 1 "f;" "6#%\"fG" }{TEXT 327 2 ", " }}{PARA 3 "" 0 "" {TEXT 354 15 " then what is " }{XPPEDIT 328 1 "f(t);" "6#-%\"fG6#%\"t G" }{TEXT 329 2 " ?" }}{PARA 3 "" 0 "" {TEXT 330 3 "a) " }{XPPEDIT 331 1 "exp(-t)*(2*cos(t)+sin(t));" "6#*&-%$expG6#,$%\"tG!\"\"\"\"\",&* &\"\"#F*-%$cosG6#F(F*F*-%$sinG6#F(F*F*" }{TEXT 332 13 " b) " }{XPPEDIT 333 1 "exp(-t)*(2*cos(2*t)+sin(2*t));" "6#*&-%$expG6#,$%\"tG !\"\"\"\"\",&*&\"\"#F*-%$cosG6#*&F-F*F(F*F*F*-%$sinG6#*&F-F*F(F*F*F*" }{TEXT 334 4 " " }}{PARA 3 "" 0 "" {TEXT 335 3 "c) " }{XPPEDIT 336 1 "exp(t)*(2*cos(t)+sin(t));" "6#*&-%$expG6#%\"tG\"\"\",&*&\"\"#F(-%$c osG6#F'F(F(-%$sinG6#F'F(F(" }{TEXT 337 18 " d) " } {XPPEDIT 338 1 "exp(t)*(2*cos(2*t)+sin(2*t));" "6#*&-%$expG6#%\"tG\"\" \",&*&\"\"#F(-%$cosG6#*&F+F(F'F(F(F(-%$sinG6#*&F+F(F'F(F(F(" }{TEXT 339 4 " " }}{PARA 3 "" 0 "" {TEXT 340 4 "e) " }{XPPEDIT 341 1 "exp (-2*t)*(2*cos(t)+sin(t));" "6#*&-%$expG6#,$*&\"\"#\"\"\"%\"tGF*!\"\"F* ,&*&F)F*-%$cosG6#F+F*F*-%$sinG6#F+F*F*" }{TEXT 342 10 " f) " } {XPPEDIT 343 1 "exp(-2*t)*(2*cos(t)+5*sin(t));" "6#*&-%$expG6#,$*&\"\" #\"\"\"%\"tGF*!\"\"F*,&*&F)F*-%$cosG6#F+F*F**&\"\"&F*-%$sinG6#F+F*F*F* " }{TEXT 344 4 " " }}{PARA 3 "" 0 "" {TEXT 345 3 "g) " }{XPPEDIT 346 1 "exp(2*t)*(2*cos(t)+sin(t));" "6#*&-%$expG6#*&\"\"#\"\"\"%\"tGF) F),&*&F(F)-%$cosG6#F*F)F)-%$sinG6#F*F)F)" }{TEXT 347 12 " h) \+ " }{XPPEDIT 348 1 "exp(2*t)*(2*cos(t)+5*sin(t));" "6#*&-%$expG6#*&\"\" #\"\"\"%\"tGF)F),&*&F(F)-%$cosG6#F*F)F)*&\"\"&F)-%$sinG6#F*F)F)F)" } {TEXT 349 4 " " }}{PARA 3 "" 0 "" {TEXT 350 3 "i) " }{XPPEDIT 351 1 "exp(-2*t)*(2*cos(2*t)+sin(2*t));" "6#*&-%$expG6#,$*&\"\"#\"\"\"%\"t GF*!\"\"F*,&*&F)F*-%$cosG6#*&F)F*F+F*F*F*-%$sinG6#*&F)F*F+F*F*F*" } {TEXT 352 6 " j) " }{XPPEDIT 353 1 "exp(2*t)*(2*cos(2*t)+sin(2*t)); " "6#*&-%$expG6#*&\"\"#\"\"\"%\"tGF)F),&*&F(F)-%$cosG6#*&F(F)F*F)F)F)- %$sinG6#*&F(F)F*F)F)F)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 692 3 "(f)" }{TEXT -1 1 "\n " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "invlaplace((2*s+9)/(s^2+ 4*s+5),s,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&-%$expG6#,$%\"tG! \"#\"\"\"-%$cosG6#F)F+\"\"#*(\"\"&F+F%F+-%$sinGF.F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 474 45 "13. Suppose that a differentiable function " } {XPPEDIT 475 0 "x;" "6#%\"xG" }{TEXT 476 13 " satisfies " }{XPPEDIT 477 0 "x*`(`*0*`)` = 7;" "6#/**%\"xG\"\"\"%\"(GF&\"\"!F&%\")GF&\"\"(" }{TEXT 478 7 ". If " }{XPPEDIT 479 0 "X;" "6#%\"XG" }{TEXT 480 31 " \+ is the Laplace transform of " }{XPPEDIT 481 0 "x;" "6#%\"xG" }{TEXT 482 12 " and if " }{XPPEDIT 483 0 "X(3) = 4;" "6#/-%\"XG6#\"\"$\" \"%" }{TEXT 484 45 ", then what is the Laplace transform of " } {XPPEDIT 485 0 "x*`'`;" "6#*&%\"xG\"\"\"%\"'GF%" }{TEXT 486 96 " eva luated at 3? (Assume that 3 is in the domain of both X and the Lap lace transform of " }{XPPEDIT 487 0 "x*`'`;" "6#*&%\"xG\"\"\"%\"'GF% " }{TEXT 488 4 " .)\n" }}{PARA 3 "" 0 "" {TEXT 509 158 "a) -1 b) \+ 0 c) 1 d) 2 e) 3 f) 4 g) 5 h) 6 \+ i) 7 \n\nj) Not enough information has been given to determine the v alue. " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 701 3 "(g)" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "laplace( d iff(x(t),t),t,s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&%#s|irG\"\"\" -%(laplaceG6%-%\"xG6#%\"tGF-F%F&F&-F+6#\"\"!!\"\"" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 16 "subs(x(0)=7, %);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&%#s|irG\"\"\"-%(laplaceG6%-%\"xG6#%\"tGF-F%F&F&\"\" (!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "subs(laplace(x(t) ,t,s) = X(s), %);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&%#s|irG\"\"\" -%\"XG6#F%F&F&\"\"(!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "subs(s = 3, %);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%\"XG6#\"\"$F' \"\"(!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "subs(X(3)=4, \+ %);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 490 44 "14. Suppose that a di fferentiable function " }{XPPEDIT 491 0 "x;" "6#%\"xG" }{TEXT 492 13 " satisfies " }{XPPEDIT 493 0 "x(0) = 8;" "6#/-%\"xG6#\"\"!\"\")" } {TEXT 494 6 " and " }{XPPEDIT 495 0 "D(x)(0) = -1;" "6#/--%\"DG6#%\"x G6#\"\"!,$\"\"\"!\"\"" }{TEXT 496 7 ". If " }{XPPEDIT 497 0 "X;" "6# %\"XG" }{TEXT 498 31 " is the Laplace transform of " }{XPPEDIT 499 0 "x;" "6#%\"xG" }{TEXT 500 12 " and if " }{XPPEDIT 501 0 "X(2) = \+ 7;" "6#/-%\"XG6#\"\"#\"\"(" }{TEXT 502 45 ", then what is the Laplac e transform of " }{XPPEDIT 503 0 "x*`''`;" "6#*&%\"xG\"\"\"%#''GF% " }{TEXT 504 99 " evaluated at 2?\n(Assume that 2 is in the domain of X and the Laplace transforms of both " }{XPPEDIT 505 0 "x*`' `;" "6#*&%\"xG\"\"\"%\"'GF%" }{TEXT 506 9 " and " }{XPPEDIT 507 0 "x*`''`;" "6#*&%\"xG\"\"\"%#''GF%" }{TEXT 508 3 ".)\n" }}{PARA 3 "" 0 "" {TEXT 489 177 "a) 24 b) 20 c) 16 d) 12 \+ e) 8 f) 4 g) 17 h) 13 i) 9\n\nj) Not e nough information has been given to determine the value. " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 702 3 "(h)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "laplace( dif f(x(t),t$2),t,s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&%#s|irG\"\"\" ,&*&F%F&-%(laplaceG6%-%\"xG6#%\"tGF/F%F&F&-F-6#\"\"!!\"\"F&F&--%\"DG6# F-F1F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "subs(\{x(0)=8,D(x )(0)= -1\}, %);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&%#s|irG\"\"\",& *&F%F&-%(laplaceG6%-%\"xG6#%\"tGF/F%F&F&\"\")!\"\"F&F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "subs(laplace(x(t),t,s) = X(s), %); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&%#s|irG\"\"\",&*&F%F&-%\"XG6#F %F&F&\"\")!\"\"F&F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 " subs(s = 2, %);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%\"XG6#\"\"#\"\" %\"#:!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "subs(X(2)=7, \+ %);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 510 8 "15. If " }{XPPEDIT 511 0 "X;" "6#%\"XG" }{TEXT 512 31 " is th e Laplace transform of " }{XPPEDIT 513 0 "x;" "6#%\"xG" }{TEXT 514 12 " and if " }{XPPEDIT 515 0 "X(s) = (48*s+57+7*s^2)/((s+3)^2*(s- 1));" "6#/-%\"XG6#%\"sG*&,(*&\"#[\"\"\"F'F,F,\"#dF,*&\"\"(F,*$F'\"\"#F ,F,F,*&,&F'F,\"\"$F,F1,&F'F,F,!\"\"F,F6" }{TEXT 516 18 ", then what \+ is " }{XPPEDIT 517 0 "x(t);" "6#-%\"xG6#%\"tG" }{TEXT 518 126 "? (Ti me-saving tip: In the partial fraction decomposition of X(s), 0 is t he numerator of the summand whose denominator is " }{XPPEDIT 519 0 "( s+3)" "6#,&%\"sG\"\"\"\"\"$F%" }{TEXT 520 3 " .)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 521 5 " a) " }{XPPEDIT 522 0 "2 *t*exp(-3*t)+5*exp(t);" "6#,&*(\"\"#\"\"\"%\"tGF&-%$expG6#,$*&\"\"$F&F 'F&!\"\"F&F&*&\"\"&F&-F)6#F'F&F&" }{TEXT 523 13 " b) " } {XPPEDIT 524 0 "2*t*exp(3*t)+5*exp(-t);" "6#,&*(\"\"#\"\"\"%\"tGF&-%$e xpG6#*&\"\"$F&F'F&F&F&*&\"\"&F&-F)6#,$F'!\"\"F&F&" }{TEXT 525 5 "\nc) \+ " }{XPPEDIT 526 0 "3*exp(-3*t)+4*t*exp(t);" "6#,&*&\"\"$\"\"\"-%$expG 6#,$*&F%F&%\"tGF&!\"\"F&F&*(\"\"%F&F,F&-F(6#F,F&F&" }{TEXT 527 14 " \+ d) " }{XPPEDIT 528 0 "3*exp(3*t)+4*t*exp(-t);" "6#,&*&\"\"$\" \"\"-%$expG6#*&F%F&%\"tGF&F&F&*(\"\"%F&F+F&-F(6#,$F+!\"\"F&F&" }{TEXT 529 6 " \ne) " }{XPPEDIT 530 0 "6*t*exp(-3*t)+7*exp(t);" "6#,&*(\"\"' \"\"\"%\"tGF&-%$expG6#,$*&\"\"$F&F'F&!\"\"F&F&*&\"\"(F&-F)6#F'F&F&" } {TEXT 531 14 " f) " }{XPPEDIT 532 0 "6*t*exp(3*t)+7*exp(-t); " "6#,&*(\"\"'\"\"\"%\"tGF&-%$expG6#*&\"\"$F&F'F&F&F&*&\"\"(F&-F)6#,$F '!\"\"F&F&" }{TEXT 533 4 "\ng) " }{XPPEDIT 534 0 "12*exp(-3*t)+7*t*exp (t);" "6#,&*&\"#7\"\"\"-%$expG6#,$*&\"\"$F&%\"tGF&!\"\"F&F&*(\"\"(F&F- F&-F(6#F-F&F&" }{TEXT 535 12 " h) " }{XPPEDIT 536 0 "12*exp(3* t)+7*t*exp(-t);" "6#,&*&\"#7\"\"\"-%$expG6#*&\"\"$F&%\"tGF&F&F&*(\"\"( F&F,F&-F(6#,$F,!\"\"F&F&" }{TEXT 537 7 " \ni) " }{XPPEDIT 538 0 "3*t *exp(-3*t)-exp(t);" "6#,&*(\"\"$\"\"\"%\"tGF&-%$expG6#,$*&F%F&F'F&!\" \"F&F&-F)6#F'F-" }{TEXT 539 17 " j) " }{XPPEDIT 256 0 "3* t*exp(3*t)+exp(-t);" "6#,&*(\"\"$\"\"\"%\"tGF&-%$expG6#*&F%F&F'F&F&F&- F)6#,$F'!\"\"F&" }{TEXT 684 8 " " }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 691 3 "(e)" }{TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "invlaplace((48*s+57+7*s^2)/((s+3)^2 *(s-1)),s,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&%\"tG\"\"\"-%$exp G6#,$F%!\"$F&\"\"'*&\"\"(F&-F(6#F%F&F&" }}}{PARA 3 "" 0 "" {TEXT 683 1 " " }{TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 540 17 "16. Supp ose that " }{XPPEDIT 541 1 "g(t) = 0;" "6#/-%\"gG6#%\"tG\"\"!" }{TEXT 542 6 " for " }{XPPEDIT 543 1 "t < 2;" "6#2%\"tG\"\"#" }{TEXT 544 7 " and " }{XPPEDIT 545 1 "g(t) = t-5;" "6#/-%\"gG6#%\"tG,&F'\"\"\"\"\" &!\"\"" }{TEXT 546 7 " for " }{XPPEDIT 547 1 "2 < t;" "6#2\"\"#%\"tG " }{TEXT 548 56 ". If F is the Laplace transform of g, then what i s " }{XPPEDIT 549 1 "F(s);" "6#-%\"FG6#%\"sG" }{TEXT 550 2 "?\n" }} {PARA 3 "" 0 "" {TEXT 551 5 "a) " }{XPPEDIT 552 1 "exp(2*s)*(1-5*s)/ (s^2);" "6#*(-%$expG6#*&\"\"#\"\"\"%\"sGF)F),&F)F)*&\"\"&F)F*F)!\"\"F) *$F*F(F." }{TEXT 553 14 " b) " }{XPPEDIT 554 1 "exp(-2*s)*(1 -5*s)/(s^2);" "6#*(-%$expG6#,$*&\"\"#\"\"\"%\"sGF*!\"\"F*,&F*F**&\"\"& F*F+F*F,F**$F+F)F," }{TEXT 555 11 " " }}{PARA 3 "" 0 "" {TEXT 556 4 "c) " }{XPPEDIT 557 1 "exp(2*s)*(1-3*s)/(s^2);" "6#*(-%$e xpG6#*&\"\"#\"\"\"%\"sGF)F),&F)F)*&\"\"$F)F*F)!\"\"F)*$F*F(F." }{TEXT 558 14 " d) " }{XPPEDIT 559 1 "exp(-2*s)*(1-3*s)/(s^2);" "6# *(-%$expG6#,$*&\"\"#\"\"\"%\"sGF*!\"\"F*,&F*F**&\"\"$F*F+F*F,F**$F+F)F ," }{TEXT 560 4 " " }}{PARA 3 "" 0 "" {TEXT 561 3 "e) " }{XPPEDIT 562 1 "exp(2*s)*(1-7*s)/(s^2);" "6#*(-%$expG6#*&\"\"#\"\"\"%\"sGF)F),& F)F)*&\"\"(F)F*F)!\"\"F)*$F*F(F." }{TEXT 563 14 " f) " } {XPPEDIT 564 1 "exp(-2*s)*(1-5*s)/(s^2);" "6#*(-%$expG6#,$*&\"\"#\"\" \"%\"sGF*!\"\"F*,&F*F**&\"\"&F*F+F*F,F**$F+F)F," }{TEXT 565 11 " \+ " }}{PARA 3 "" 0 "" {TEXT 566 3 "g) " }{XPPEDIT 567 1 "(1-5*s)/(s ^2);" "6#*&,&\"\"\"F%*&\"\"&F%%\"sGF%!\"\"F%*$F(\"\"#F)" }{TEXT 568 27 " h) " }{XPPEDIT 569 1 "(1-3*s)/(s^2);" "6#* &,&\"\"\"F%*&\"\"$F%%\"sGF%!\"\"F%*$F(\"\"#F)" }{TEXT 570 7 " " }}{PARA 3 "" 0 "" {TEXT 571 4 "i) " }{XPPEDIT 572 1 "(1-7*s)/(s^2);" "6#*&,&\"\"\"F%*&\"\"(F%%\"sGF%!\"\"F%*$F(\"\"#F)" }{TEXT 573 75 " \+ j) Since g is not continuous, F does not exist \+ \n" }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 690 3 "(d)" }{TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "Int( (t-5)*exp(-s*t), t = 2 .. infinity);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&,&%\"tG\"\"\"\"\"&!\"\"F)-% $expG6#,$*&%\"sGF)F(F)F+F)/F(;\"\"#%)infinityG" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 9 "value(%);" }}{PARA 6 "" 1 "" {TEXT -1 68 "Defin ite integration: Can't determine if the integral is convergent." }} {PARA 6 "" 1 "" {TEXT -1 30 "Need to know the sign of --> s" }}{PARA 6 "" 1 "" {TEXT -1 57 "Will now try indefinite integration and then ta ke limits." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&limitG6$,$*&,,*(-%$ex pG6#,$*&%\"sG\"\"\"%\"tGF0!\"\"F0F/F0F1F0F0F*F0*(\"\"&F0F*F0F/F0F2-F+6 #,$F/!\"#F2*(\"\"$F0F5F0F/F0F0F0*$)F/\"\"#F0F2F2/F1%)infinityG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "assume(s>0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&*&-%$expG6#,$%#s|irG!\"#\"\"\",&F*\"\"$F,!\"\"F,F,* $)F*\"\"#F,F/F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 " " 0 "" {TEXT -1 34 "17. Calculate the convolution (sin" }{TEXT 574 1 " *" }{TEXT -1 9 "sin)(t). " }}{PARA 3 "" 0 "" {TEXT 575 3 "a) " } {XPPEDIT 576 1 "sin(t)-t*sin(t);" "6#,&-%$sinG6#%\"tG\"\"\"*&F'F(-F%6# F'F(!\"\"" }{TEXT 577 10 " b) " }{XPPEDIT 578 1 "sin(t)+t*sin(t) " "6#,&-%$sinG6#%\"tG\"\"\"*&F'F(-F%6#F'F(F(" }{TEXT 579 2 " " }} {PARA 3 "" 0 "" {TEXT 580 3 "c) " }{XPPEDIT 581 1 "cos(t)-t*sin(t);" " 6#,&-%$cosG6#%\"tG\"\"\"*&F'F(-%$sinG6#F'F(!\"\"" }{TEXT 582 9 " \+ d) " }{XPPEDIT 583 1 "cos(t)+t*sin(t);" "6#,&-%$cosG6#%\"tG\"\"\"*&F'F (-%$sinG6#F'F(F(" }{TEXT 584 2 " " }}{PARA 3 "" 0 "" {TEXT 585 3 "e) \+ " }{XPPEDIT 586 1 "cos(t)-2*t*sin(t);" "6#,&-%$cosG6#%\"tG\"\"\"*(\"\" #F(F'F(-%$sinG6#F'F(!\"\"" }{TEXT 587 7 " f) " }{XPPEDIT 588 1 "cos (t)+2*t*sin(t);" "6#,&-%$cosG6#%\"tG\"\"\"*(\"\"#F(F'F(-%$sinG6#F'F(F( " }{TEXT 589 2 " " }}{PARA 3 "" 0 "" {TEXT 590 3 "g) " }{XPPEDIT 591 1 "sin(t)/2-t*cos(t);" "6#,&*&-%$sinG6#%\"tG\"\"\"\"\"#!\"\"F)*&F(F)-% $cosG6#F(F)F+" }{TEXT 592 9 " h) " }{XPPEDIT 593 1 "sin(t)/2+t*co s(t);" "6#,&*&-%$sinG6#%\"tG\"\"\"\"\"#!\"\"F)*&F(F)-%$cosG6#F(F)F)" } {TEXT 594 1 " " }}{PARA 3 "" 0 "" {TEXT 595 3 "i) " }{XPPEDIT 596 1 "s in(t)/2-t*cos(t)/2;" "6#,&*&-%$sinG6#%\"tG\"\"\"\"\"#!\"\"F)*(F(F)-%$c osG6#F(F)F*F+F+" }{TEXT 597 9 " j) " }{XPPEDIT 598 1 "sin(t)/2+t* cos(t)/2;" "6#,&*&-%$sinG6#%\"tG\"\"\"\"\"#!\"\"F)*(F(F)-%$cosG6#F(F)F *F+F)" }{TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 689 3 "(i)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "int(sin(t au)*sin(t-tau),tau=0..t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&-%$co sG6#%\"tG\"\"\"F(F)#!\"\"\"\"#*&#F)F,F)-%$sinGF'F)F)" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 4 "18. " }{TEXT 599 6 " Let " }{XPPEDIT 600 1 "f;" "6#%\"fG" }{TEXT 601 98 " be the periodic function with period 2 that is defined on the interval [0,2) by the formula " } {XPPEDIT 602 0 "f(t) = PIECEWISE([1, 0 <= t and t < 1],[0, 1 <= t and \+ t < 2]);" "6#/-%\"fG6#%\"tG-%*PIECEWISEG6$7$\"\"\"31\"\"!F'2F'F,7$F/31 F,F'2F'\"\"#" }{TEXT 603 35 ".\nWhat is the Laplace transform of " } {XPPEDIT 604 1 "f;" "6#%\"fG" }{TEXT 605 2 " ?" }}{PARA 0 "" 0 "" {TEXT 606 0 "" }}{PARA 3 "" 0 "" {TEXT 607 3 "a) " }{XPPEDIT 608 0 "(1 +exp(-s))/((1-exp(-2*s))*s);" "6#*&,&\"\"\"F%-%$expG6#,$%\"sG!\"\"F%F% *&,&F%F%-F'6#,$*&\"\"#F%F*F%F+F+F%F*F%F+" }{TEXT 609 11 " b) " }{XPPEDIT 610 0 "(1-exp(-s))/((1-exp(-2*s))*s);" "6#*&,&\"\"\"F%-%$exp G6#,$%\"sG!\"\"F+F%*&,&F%F%-F'6#,$*&\"\"#F%F*F%F+F+F%F*F%F+" }{TEXT 611 11 " c) " }{XPPEDIT 612 0 "(1-exp(-s))/(1-exp(-2*s));" "6#* &,&\"\"\"F%-%$expG6#,$%\"sG!\"\"F+F%,&F%F%-F'6#,$*&\"\"#F%F*F%F+F+F+" }{TEXT 613 5 " " }}{PARA 3 "" 0 "" {TEXT 614 3 "d) " }{XPPEDIT 615 0 "(1+exp(-s))/(1-exp(-2*s));" "6#*&,&\"\"\"F%-%$expG6#,$%\"sG!\" \"F%F%,&F%F%-F'6#,$*&\"\"#F%F*F%F+F+F+" }{TEXT 616 18 " \+ e) " }{XPPEDIT 617 0 "exp(-s)/(1-exp(-2*s));" "6#*&-%$expG6#,$%\"sG!\" \"\"\"\",&F*F*-F%6#,$*&\"\"#F*F(F*F)F)F)" }{TEXT 618 17 " \+ f) " }{XPPEDIT 619 0 "s/(1-exp(-2*s));" "6#*&%\"sG\"\"\",&F%F%-%$expG 6#,$*&\"\"#F%F$F%!\"\"F-F-" }{TEXT 620 4 " " }}{PARA 0 "" 0 "" {TEXT 621 3 "g) " }{XPPEDIT 622 0 "s^2/(1-exp(-2*s));" "6#*&%\"sG\"\"# ,&\"\"\"F'-%$expG6#,$*&F%F'F$F'!\"\"F-F-" }{TEXT 623 18 " \+ h) " }{XPPEDIT 624 0 "(1-s)/(1-exp(-2*s));" "6#*&,&\"\"\"F%%\"sG!\" \"F%,&F%F%-%$expG6#,$*&\"\"#F%F&F%F'F'F'" }{TEXT 625 7 " \ni) " } {XPPEDIT 626 0 "(1+s)/(1-exp(-2*s));" "6#*&,&\"\"\"F%%\"sGF%F%,&F%F%-% $expG6#,$*&\"\"#F%F&F%!\"\"F.F." }{TEXT 627 20 " j) " }{XPPEDIT 628 0 "1/((1-exp(-2*s))*(s+1));" "6#*&\"\"\"F$*&,&F$F$-%$exp G6#,$*&\"\"#F$%\"sGF$!\"\"F.F$,&F-F$F$F$F$F." }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 686 3 "(b)" }{TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "1/( 1-exp(-2*s))*int(exp(-s*t),t=0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #,$*&,&-%$expG6#,$%\"sG!\"\"\"\"\"F,F+F,*&,&F,F,-F'6#,$F*!\"#F+F,F*F,F +F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 659 30 "19. The Laplace transform of " }{XPPEDIT 660 1 "f(t) = sin(t)/t; " "6#/-%\"fG6#%\"tG*&-%$sinG6#F'\"\"\"F'!\"\"" }{TEXT 661 6 " is " } {XPPEDIT 662 1 "F(s) = arctan(1/s);" "6#/-%\"FG6#%\"sG-%'arctanG6#*&\" \"\"F,F'!\"\"" }{TEXT 663 57 ". If the Laplace transform of x(t) is F' (s) then what is " }{XPPEDIT 664 0 "x(Pi/6);" "6#-%\"xG6#*&%#PiG\"\"\" \"\"'!\"\"" }{TEXT 665 3 "? " }}{PARA 3 "" 0 "" {TEXT 666 31 "a) 2 \+ b) 1/2 c) " }{XPPEDIT 667 1 "sqrt(3)/2;" "6#*&-%%sqrtG6# \"\"$\"\"\"\"\"#!\"\"" }{TEXT 668 13 " d) " }{XPPEDIT 669 1 " sqrt(2);" "6#-%%sqrtG6#\"\"#" }{TEXT 670 13 " e) " }{XPPEDIT 671 1 "sqrt(2)/2;" "6#*&-%%sqrtG6#\"\"#\"\"\"F'!\"\"" }{TEXT 672 1 " \+ " }}{PARA 3 "" 0 "" {TEXT 673 32 "f) -2 g) -1/2 h) " } {XPPEDIT 674 1 "-sqrt(3)/2;" "6#,$*&-%%sqrtG6#\"\"$\"\"\"\"\"#!\"\"F+ " }{TEXT 675 9 " i) " }{XPPEDIT 676 1 "-sqrt(2);" "6#,$-%%sqrtG6# \"\"#!\"\"" }{TEXT 677 11 " j) " }{XPPEDIT 678 1 "-sqrt(2)/2;" "6#,$*&-%%sqrtG6#\"\"#\"\"\"F(!\"\"F*" }{TEXT 679 3 " " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 688 3 "(g)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "laplace( -t* g(t),t,s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%diffG6$-%(laplaceG6%- %\"gG6#%\"tGF,%\"sGF-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "x( t) = -t*(sin(t)/t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"xG6#%\"tG, $-%$sinGF&!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "subs(t=P i/6, %);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"xG6#,$%#PiG#\"\"\"\" \"',$-%$sinGF&!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value (%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"xG6#,$%#PiG#\"\"\"\"\"'#! \"\"\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 " " {TEXT 630 18 "20. Suppose that " }{XPPEDIT 256 0 "x(t);" "6#-%\"xG6 #%\"tG" }{TEXT 629 1 " " }{TEXT 631 45 " is the solution of the initia l value problem" }{XPPEDIT 632 1 "diff(x(t),`$`(t,2))+4*x(t) = delta[2 ](t),x(0) = 0,D(x)(0) = 1;" "6%/,&-%%diffG6$-%\"xG6#%\"tG-%\"$G6$F+\" \"#\"\"\"*&\"\"%F0-F)6#F+F0F0-&%&deltaG6#F/6#F+/-F)6#\"\"!F=/--%\"DG6# F)6#F=F0" }{TEXT 633 33 " . What is the Laplace transform " }{XPPEDIT 634 1 "X(s);" "6#-%\"XG6#%\"sG" }{TEXT 635 4 " of " }{XPPEDIT 636 1 "x (t);" "6#-%\"xG6#%\"tG" }{TEXT 637 1 "?" }}{PARA 3 "" 0 "" {TEXT 638 3 "a) " }{XPPEDIT 639 0 "delta[2](s)/(s^2+4);" "6#*&-&%&deltaG6#\"\"#6 #%\"sG\"\"\",&*$F*F(F+\"\"%F+!\"\"" }{TEXT 640 7 " b) " }{XPPEDIT 641 0 "2*delta[2](s)/(s^2+4);" "6#*(\"\"#\"\"\"-&%&deltaG6#F$6#%\"sGF% ,&*$F+F$F%\"\"%F%!\"\"" }{TEXT 642 7 " c) " }{XPPEDIT 643 0 "4*delt a[2](s)/(s^2+4);" "6#*(\"\"%\"\"\"-&%&deltaG6#\"\"#6#%\"sGF%,&*$F,F*F% F$F%!\"\"" }{TEXT 644 7 " d) " }{XPPEDIT 645 0 "(2+delta[2](s))/(s^ 2+4);" "6#*&,&\"\"#\"\"\"-&%&deltaG6#F%6#%\"sGF&F&,&*$F,F%F&\"\"%F&!\" \"" }{TEXT 646 6 " e) " }{XPPEDIT 647 0 "2*exp(-2*s)/(s^2+4);" "6#*( \"\"#\"\"\"-%$expG6#,$*&F$F%%\"sGF%!\"\"F%,&*$F+F$F%\"\"%F%F," }{TEXT 648 6 " \nf) " }{XPPEDIT 649 0 "2*exp(2*s)/(s^2+4);" "6#*(\"\"#\"\"\" -%$expG6#*&F$F%%\"sGF%F%,&*$F*F$F%\"\"%F%!\"\"" }{TEXT 650 7 " g) \+ " }{XPPEDIT 651 0 "(1+exp(-2*s))/(s^2+4);" "6#*&,&\"\"\"F%-%$expG6#,$* &\"\"#F%%\"sGF%!\"\"F%F%,&*$F,F+F%\"\"%F%F-" }{TEXT 652 7 " h) " } {XPPEDIT 653 0 "(1+exp(2*s))/(s^2+4);" "6#*&,&\"\"\"F%-%$expG6#*&\"\"# F%%\"sGF%F%F%,&*$F+F*F%\"\"%F%!\"\"" }{TEXT 654 8 " i) " } {XPPEDIT 655 0 "exp(-2*s)/(s^2+4);" "6#*&-%$expG6#,$*&\"\"#\"\"\"%\"sG F*!\"\"F*,&*$F+F)F*\"\"%F*F," }{TEXT 656 10 " j) " }{XPPEDIT 657 0 "exp(2*s)/(s^2+4);" "6#*&-%$expG6#*&\"\"#\"\"\"%\"sGF)F),&*$F*F( F)\"\"%F)!\"\"" }{TEXT 658 1 " " }}{PARA 3 "" 0 "" {TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT 687 3 "(g)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "dsolve( \{diff(x(t),t$2)+4*x (t) = Dirac(t-2), x(0) = 0, D(x)(0) = 1\},x(t),method=laplace);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"xG6#%\"tG,&-%$sinG6#,$F'\"\"##\" \"\"F-**#F/\"\"%F/-%*HeavisideG6#,&F'F/F-!\"\"F/-%%sqrtG6#F2F/-F*6#*&F 8F/F6F/F/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "laplace( rhs (%),t,s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"\"\"F%,&*$)%\"sG\" \"#F%F%\"\"%F%!\"\"F%*&*(#F%\"\")F%-%%sqrtG6#F+F%-%$expG6#,$F)!\"#F%F% ,&F'#F%F+F%F%F,F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "normal (%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,&\"\"#\"\"\"*&-%%sqrtG6# \"\"%F'-%$expG6#,$%\"sG!\"#F'F'F',&*$)F1F&F'F'F,F'!\"\"#F'F&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&\"\"\"F%-%$expG6#,$%\"sG!\"#F%F%,&*$)F*\"\"#F% F%\"\"%F%!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 "Or" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "map(z->lapla ce(z,t,s),diff(x(t),t$2)+4*x(t) = Dirac(t-2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&%\"sG\"\"\",&*&F&F'-%(laplaceG6%-%\"xG6#%\"tGF0F&F 'F'-F.6#\"\"!!\"\"F'F'--%\"DG6#F.F2F4*&\"\"%F'F*F'F'-%$expG6#,$F&!\"# " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "subs(\{x(0)=0, D(x)(0) \+ = 0\},%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&)%\"sG\"\"#\"\"\"-%( laplaceG6%-%\"xG6#%\"tGF0F'F)F)*&\"\"%F)F*F)F)-%$expG6#,$F'!\"#" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "solve(%, laplace(x(t),t,s)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%$expG6#,$%\"sG!\"#\"\"\",&*$)F (\"\"#F*F*\"\"%F*!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{MARK "10 0 1" 3 }{VIEWOPTS 1 0 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }