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}{CSTYLE "" -1 845 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" 19 846 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 847 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 848 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 849 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 850 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" 19 851 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 852 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 854 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 855 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 856 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" 19 857 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 858 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 859 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 860 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 861 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" 19 862 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 863 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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{CSTYLE "" 19 882 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 883 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 884 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 885 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 886 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 887 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 888 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 889 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 890 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 891 "" 1 12 0 0 0 0 0 0 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 "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courier " 1 10 0 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Title" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Normal" -1 256 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 "Heading 1" -1 257 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 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 "T imes" 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 } } {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT 804 30 " Math 217 Fall 2000 Fina l Exam" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 268 46 "Notational Remark: In this exam, the symbol " }{XPPEDIT 19 1 "di ff(y(x),x)" "6#-%%diffG6$-%\"yG6#%\"xGF)" }{TEXT 270 9 " means " } {XPPEDIT 18 0 "dy/dx;" "6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT 269 122 ". \+ \nNo calculators of any kind. No \"cheat sheets.\" A brief table of La place transforms may be found at the end of the exam." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 282 75 "1. Which one of the differential equations listed in answers (a) - (h) is " } {TEXT 272 3 "not" }{TEXT 283 179 " exact? (If each of the equations i n answers (a)-(h) is exact, then choose answer (j). If two or more of \+ the equations in answers (a)-(h) are not exact, then choose answer (i) . )" }{TEXT -1 2 "\n\n" }{TEXT 284 6 "a) " }{XPPEDIT 256 1 "2*x*y^3 *dx+3*x^2*y^2*dy = 0;" "6#/,&**\"\"#\"\"\"%\"xGF'%\"yG\"\"$%#dxGF'F'** F*F'*$F(F&F'F)F&%#dyGF'F'\"\"!" }{TEXT 274 7 "\n\nb) " }{XPPEDIT 258 1 "cos(x+y)*dx+cos(x+y)*dy = 0;" "6#/,&*&-%$cosG6#,&%\"xG\"\"\"%\" yGF+F+%#dxGF+F+*&-F'6#,&F*F+F,F+F+%#dyGF+F+\"\"!" }{TEXT 275 7 "\n\nc) " }{XPPEDIT 259 1 "y*e^(x*y)*dx+x*e^(x*y)*dy = 0;" "6#/,&*(%\"yG\" \"\")%\"eG*&%\"xGF'F&F'F'%#dxGF'F'*(F+F')F)*&F+F'F&F'F'%#dyGF'F'\"\"! " }{TEXT 276 6 "\n\nd) " }{XPPEDIT 260 1 "x*e^(x*y)*dx+y*e^(x*y)*dy = 0;" "6#/,&*(%\"xG\"\"\")%\"eG*&F&F'%\"yGF'F'%#dxGF'F'*(F+F')F)*&F&F'F +F'F'%#dyGF'F'\"\"!" }{TEXT 277 5 "\n\ne) " }{XPPEDIT 261 1 "(2*x+3*y) *dx+(3*x+2)*dy = 0;" "6#/,&*&,&*&\"\"#\"\"\"%\"xGF)F)*&\"\"$F)%\"yGF)F )F)%#dxGF)F)*&,&*&F,F)F*F)F)F(F)F)%#dyGF)F)\"\"!" }{TEXT 278 5 "\n\nf) " }{XPPEDIT 262 1 "(4*x-y)*dx+(6*y-x)*dy = 0;" "6#/,&*&,&*&\"\"%\"\" \"%\"xGF)F)%\"yG!\"\"F)%#dxGF)F)*&,&*&\"\"'F)F+F)F)F*F,F)%#dyGF)F)\"\" !" }{TEXT 279 5 "\n\ng) " }{XPPEDIT 263 1 "(2*x*y^2+3*x^2)*dx+(2*x^2*y +4*y^3)*dy = 0;" "6#/,&*&,&*(\"\"#\"\"\"%\"xGF)%\"yGF(F)*&\"\"$F)*$F*F (F)F)F)%#dxGF)F)*&,&*(F(F)*$F*F(F)F+F)F)*&\"\"%F)*$F+F-F)F)F)%#dyGF)F) \"\"!" }{TEXT 280 5 "\n\nh) " }{XPPEDIT 264 1 "(3*x^2+2*y^2)*dx+(4*x*y +J[0](y))*dy = 0;" "6#/,&*&,&*&\"\"$\"\"\"*$%\"xG\"\"#F)F)*&F,F)*$%\"y GF,F)F)F)%#dxGF)F)*&,&*(\"\"%F)F+F)F/F)F)-&%\"JG6#\"\"!6#F/F)F)%#dyGF) F)F9" }{TEXT 281 124 "\n\ni) At least two of the differential equatio ns above are not exact.\n\nj) All of the above differential equations \+ are exact." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 709 13 "Solution: (d)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 710 48 "All equations here ha ve been written in the form" }{TEXT -1 2 " " }{XPPEDIT 19 1 "M(x,y)*d x+N(x,y)*dy = 0;" "6#/,&*&-%\"MG6$%\"xG%\"yG\"\"\"%#dxGF+F+*&-%\"NG6$F )F*F+%#dyGF+F+\"\"!" }{TEXT -1 2 ". " }{TEXT 711 59 "We must therefore determine which equation fails to satisfy" }{TEXT -1 2 " " } {XPPEDIT 19 1 "diff(M(x,y),y) <> diff(N(x,y),x);" "6#0-%%diffG6$-%\"MG 6$%\"xG%\"yGF+-F%6$-%\"NG6$F*F+F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "diff(2*x*y^3 ,y) = diff(3*x^2*y^2, x); evalb(%); # a exact" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*&%\"xG\"\"\")%\"yG\"\"#F'\"\"'F$" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "diff(cos(x+y),y) = diff(cos(x+y), x); evalb(%); # b exact" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$-%$sinG6#,&%\"xG\"\"\"%\"yGF*!\"\"F$" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "diff(y*e^(x*y),y) = diff(x*e^(x*y), x); evalb(%); # c exact" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&)%\"eG*&%\"xG\"\"\"%\"yGF )F)**F*F)F%F)F(F)-%#lnG6#F&F)F)F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#% %trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "diff(x*e^(x*y),y) = diff(y*e^(x*y), x); evalb(%); # d not exact" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*()%\"xG\"\"#\"\"\")%\"eG*&F&F(%\"yGF(F(-%#lnG6#F*F(*( )F,F'F(F)F(F-F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "diff((2*x+3*y),y) = diff((3* x+2), x); evalb(%); # e exact" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\" $F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 55 "diff((4*x-y),y) = diff((6*y-x), x); evalb(%); \+ # f exact" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/!\"\"F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "diff((2*x*y^2+3*x^2),y) = diff((2*x^2*y+4*y^3), x); evalb(%); # g \+ exact" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*&%\"xG\"\"\"%\"yGF'\"\"%F $" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "diff((3*x^2+2*y^2),y) = diff((4*x*y+J[0](y)), x) ; evalb(%); # h exact" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$%\"yG\"\"% F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 712 39 "Only the equation in ans wer (d) is not " }{TEXT -1 6 "exact." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 319 45 "2. When applied to the differ ential equation " }{XPPEDIT 320 1 "diff(y(x),x)+f(x)*y(x) = y(x)^5;" " 6#/,&-%%diffG6$-%\"yG6#%\"xGF+\"\"\"*&-%\"fG6#F+F,-F)6#F+F,F,*$-F)6#F+ \"\"&" }{TEXT 321 91 " which of the following changes of variable resu lts in a linear differential equation?\n\na) " }{XPPEDIT 322 1 "w(x) = f(x)*y(x);" "6#/-%\"wG6#%\"xG*&-%\"fG6#F'\"\"\"-%\"yG6#F'F," }{TEXT 310 4 "\nb) " }{XPPEDIT 323 1 "w(x) = f(x)*y(x)^5;" "6#/-%\"wG6#%\"xG* &-%\"fG6#F'\"\"\"*$-%\"yG6#F'\"\"&F," }{TEXT 311 4 "\nc) " }{XPPEDIT 324 1 "w(x) = f(x)^5*y(x);" "6#/-%\"wG6#%\"xG*&-%\"fG6#F'\"\"&-%\"yG6# F'\"\"\"" }}{PARA 3 "" 0 "" {TEXT 313 3 "d) " }{XPPEDIT 325 1 "w(x) = \+ x^5*y(x);" "6#/-%\"wG6#%\"xG*&F'\"\"&-%\"yG6#F'\"\"\"" }}{PARA 3 "" 0 "" {TEXT 314 3 "e) " }{XPPEDIT 326 1 "w(x) = y(x)*exp(Int(f(x),x));" " 6#/-%\"wG6#%\"xG*&-%\"yG6#F'\"\"\"-%$expG6#-%$IntG6$-%\"fG6#F'F'F," }} {PARA 3 "" 0 "" {TEXT 315 4 "f) " }{XPPEDIT 327 1 "w(x) = y(x)*exp(-I nt(f(x),x));" "6#/-%\"wG6#%\"xG*&-%\"yG6#F'\"\"\"-%$expG6#,$-%$IntG6$- %\"fG6#F'F'!\"\"F," }}{PARA 3 "" 0 "" {TEXT 316 3 "g) " }{XPPEDIT 328 1 "w(x) = y(x)^5;" "6#/-%\"wG6#%\"xG*$-%\"yG6#F'\"\"&" }}{PARA 3 "" 0 "" {TEXT 317 3 "h) " }{XPPEDIT 329 1 "w(x) = 1/(y(x)^5);" "6#/-%\"wG6# %\"xG*&\"\"\"F)*$-%\"yG6#F'\"\"&!\"\"" }}{PARA 3 "" 0 "" {TEXT 318 3 " i) " }{XPPEDIT 330 1 "w(x) = y(x)^4;" "6#/-%\"wG6#%\"xG*$-%\"yG6#F'\" \"%" }{TEXT 312 4 "\nj) " }{XPPEDIT 273 1 "w(x) = 1/(y(x)^4);" "6#/-% \"wG6#%\"xG*&\"\"\"F)*$-%\"yG6#F'\"\"%!\"\"" }}{PARA 3 "" 0 "" {TEXT 805 0 "" }}{PARA 0 "" 0 "" {TEXT 713 9 "Solution:" }{TEXT -1 2 " " } {TEXT 714 5 "( j )" }{TEXT 716 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT 715 34 "To convert the Bernoulii equation " } {XPPEDIT 19 1 "diff(y(x),x)+f(x)*y(x) = y(x)^n;" "6#/,&-%%diffG6$-%\"y G6#%\"xGF+\"\"\"*&-%\"fG6#F+F,-F)6#F+F,F,)-F)6#F+%\"nG" }{TEXT 717 53 " into a linear equation, make the change of variable " }{XPPEDIT 19 1 "w(x) = y(x)^(1-n);" "6#/-%\"wG6#%\"xG)-%\"yG6#F',&\"\"\"F-%\"nG!\" \"" }{TEXT 718 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 331 46 "3. Which one of the following expressions for " }{XPPEDIT 257 1 "y(x);" "6#-%\"yG6 #%\"xG" }{TEXT 332 44 " is a solution to the differential equation " } {XPPEDIT 333 1 "diff(y(x),`$`(x,2))-4*diff(y(x),x)+3*y(x) = 0;" "6#/,( -%%diffG6$-%\"yG6#%\"xG-%\"$G6$F+\"\"#\"\"\"*&\"\"%F0-F&6$-F)6#F+F+F0! \"\"*&\"\"$F0-F)6#F+F0F0\"\"!" }{TEXT 334 4 "? \n" }}{PARA 257 "" 0 " " {TEXT 806 3 "a) " }{XPPEDIT 807 1 "2*exp(x)+exp(-3*x);" "6#,&*&\"\"# \"\"\"-%$expG6#%\"xGF&F&-F(6#,$*&\"\"$F&F*F&!\"\"F&" }{TEXT 808 22 " \+ b) " }{XPPEDIT 809 1 "exp(-x)+2*exp(3*x);" "6#,&-%$ex pG6#,$%\"xG!\"\"\"\"\"*&\"\"#F*-F%6#*&\"\"$F*F(F*F*F*" }{TEXT 810 17 " c) " }{XPPEDIT 811 1 "5*exp(x)-exp(3*x);" "6#,&*&\"\"&\" \"\"-%$expG6#%\"xGF&F&-F(6#*&\"\"$F&F*F&!\"\"" }{TEXT 812 2 " " }} {PARA 257 "" 0 "" {TEXT 813 3 "d) " }{XPPEDIT 814 1 "exp(-x)-exp(-3*x) ;" "6#,&-%$expG6#,$%\"xG!\"\"\"\"\"-F%6#,$*&\"\"$F*F(F*F)F)" }{TEXT 815 19 " e) " }{XPPEDIT 816 1 "exp(x)+x*exp(x);" "6#,&- %$expG6#%\"xG\"\"\"*&F'F(-F%6#F'F(F(" }{TEXT 817 30 " \+ f) " }{XPPEDIT 818 1 "exp(2*x)+2*x*exp(2*x);" "6#,&-%$expG6# *&\"\"#\"\"\"%\"xGF)F)*(F(F)F*F)-F%6#*&F(F)F*F)F)F)" }{TEXT 819 2 " \+ " }}{PARA 257 "" 0 "" {TEXT 820 3 "g) " }{XPPEDIT 821 1 "exp(-4*x)*sin (3*x);" "6#*&-%$expG6#,$*&\"\"%\"\"\"%\"xGF*!\"\"F*-%$sinG6#*&\"\"$F*F +F*F*" }{TEXT 822 20 " h) " }{XPPEDIT 823 1 "exp(4*x)* sin(3*x);" "6#*&-%$expG6#*&\"\"%\"\"\"%\"xGF)F)-%$sinG6#*&\"\"$F)F*F)F )" }{TEXT 824 21 " i) " }{XPPEDIT 825 1 "exp(2*x)*sin (sqrt(3)*x);" "6#*&-%$expG6#*&\"\"#\"\"\"%\"xGF)F)-%$sinG6#*&-%%sqrtG6 #\"\"$F)F*F)F)" }{TEXT 826 2 " " }}{PARA 257 "" 0 "" {TEXT 827 3 "j) \+ " }{XPPEDIT 828 1 "exp((-2)*x)*sin(sqrt(3)*x);" "6#*&-%$expG6#*&,$\"\" #!\"\"\"\"\"%\"xGF+F+-%$sinG6#*&-%%sqrtG6#\"\"$F+F,F+F+" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 743 9 "Solution:" }{TEXT -1 2 " " }{TEXT 744 5 "( c )" }{TEXT 746 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 745 0 "" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "dsolve(diff(y(x), x$2)- 4*diff(y(x),x)+3*y(x) = 0, y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ -%\"yG6#%\"xG,&*&%$_C1G\"\"\"-%$expGF&F+F+*&%$_C2GF+-F-6#,$F'\"\"$F+F+ " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 285 40 "4. Consider the differential equation\n \n" }{XPPEDIT 286 1 "d iff(y(x),x)+3*y(x)/x = cos(x^4);" "6#/,&-%%diffG6$-%\"yG6#%\"xGF+\"\" \"*(\"\"$F,-F)6#F+F,F+!\"\"F,-%$cosG6#*$F+\"\"%" }{TEXT 287 56 ". \n\n What is a sensible first step towards its solution?\n" }}{PARA 3 "" 0 "" {TEXT 288 32 "a) Make the change of variable " }{XPPEDIT 289 1 "v( x) = y(x)/x;" "6#/-%\"vG6#%\"xG*&-%\"yG6#F'\"\"\"F'!\"\"" }{TEXT 290 34 ".\nb) Make the change of variable " }{XPPEDIT 291 1 "v(x) = y(x)^ 3;" "6#/-%\"vG6#%\"xG*$-%\"yG6#F'\"\"$" }{TEXT 292 34 ".\nc) Make the \+ change of variable " }{XPPEDIT 293 1 "v(x) = 1/(y(x)^3);" "6#/-%\"vG6 #%\"xG*&\"\"\"F)*$-%\"yG6#F'\"\"$!\"\"" }{TEXT 294 1 "." }}{PARA 3 "" 0 "" {TEXT 295 42 "d) Multiply both sides of the equation by " } {XPPEDIT 296 1 "x;" "6#%\"xG" }{TEXT 297 44 ".\ne) Multiply both sides of the equation by " }{XPPEDIT 298 1 "x/3;" "6#*&%\"xG\"\"\"\"\"$!\" \"" }{TEXT 299 44 ".\nf) Multiply both sides of the equation by " } {XPPEDIT 300 1 "x^2;" "6#*$%\"xG\"\"#" }{TEXT 301 44 ".\ng) Multiply b oth sides of the equation by " }{XPPEDIT 302 1 "x^3;" "6#*$%\"xG\"\"$ " }{TEXT 303 44 ".\nh) Multiply both sides of the equation by " } {XPPEDIT 304 1 "x^4;" "6#*$%\"xG\"\"%" }{TEXT 305 44 ".\ni) Multiply b oth sides of the equation by " }{XPPEDIT 306 1 "sec(x^4);" "6#-%$secG6 #*$%\"xG\"\"%" }{TEXT 307 16 ".\nj) Substitute " }{XPPEDIT 308 1 "y(x) = C*cos(x^4);" "6#/-%\"yG6#%\"xG*&%\"CG\"\"\"-%$cosG6#*$F'\"\"%F*" } {TEXT 309 30 " and solve for the constant C." }}{PARA 3 "" 0 "" {TEXT 271 0 "" }}{PARA 0 "" 0 "" {TEXT 747 9 "Solution:" }{TEXT -1 2 " " } {TEXT 748 5 "( g )" }{TEXT 750 1 " " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 749 114 "The equation is linear, in standard form. The first ste p toward solution is to multiply by the integrating factor " } {XPPEDIT 19 1 "exp(Int(3/x,x));" "6#-%$expG6#-%$IntG6$*&\"\"$\"\"\"%\" xG!\"\"F," }{TEXT 751 14 ", that is, by " }{XPPEDIT 19 1 "x^3;" "6#*$% \"xG\"\"$" }{TEXT 752 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 335 46 "5. Whic h one of the following expressions for " }{XPPEDIT 336 1 "y(x);" "6#-% \"yG6#%\"xG" }{TEXT 337 44 " is a solution to the differential equatio n " }{XPPEDIT 338 1 "9*diff(y(x),`$`(x,2))+36*y(x) = 0;" "6#/,&*&\"\"* \"\"\"-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F.\"\"#F'F'*&\"#OF'-F,6#F.F'F'\" \"!" }{TEXT 339 1 "?" }}{PARA 257 "" 0 "" {TEXT 829 3 "a) " }{XPPEDIT 830 1 "2*cos(2*x)-3*sin(2*x);" "6#,&*&\"\"#\"\"\"-%$cosG6#*&F%F&%\"xGF &F&F&*&\"\"$F&-%$sinG6#*&F%F&F+F&F&!\"\"" }{TEXT 831 6 " b) " } {XPPEDIT 832 1 "3*sin(2*x)+x*sin(2*x);" "6#,&*&\"\"$\"\"\"-%$sinG6#*& \"\"#F&%\"xGF&F&F&*&F,F&-F(6#*&F+F&F,F&F&F&" }{TEXT 833 10 " c) \+ " }{XPPEDIT 834 1 "4*exp(x)*(cos(x)+sin(x));" "6#*(\"\"%\"\"\"-%$expG6 #%\"xGF%,&-%$cosG6#F)F%-%$sinG6#F)F%F%" }{TEXT 835 2 " " }}{PARA 257 "" 0 "" {TEXT 836 3 "d) " }{XPPEDIT 837 1 "exp(10*x)-exp(4*x);" "6#,&- %$expG6#*&\"#5\"\"\"%\"xGF)F)-F%6#*&\"\"%F)F*F)!\"\"" }{TEXT 838 15 " \+ e) " }{XPPEDIT 839 1 "exp(5*x)+exp(10*x);" "6#,&-%$expG6#*& \"\"&\"\"\"%\"xGF)F)-F%6#*&\"#5F)F*F)F)" }{TEXT 840 20 " \+ f) " }{XPPEDIT 841 1 "cos(4*x)-3*sin(4*x);" "6#,&-%$cosG6#*&\"\"%\" \"\"%\"xGF)F)*&\"\"$F)-%$sinG6#*&F(F)F*F)F)!\"\"" }{TEXT 842 2 " " }} {PARA 257 "" 0 "" {TEXT 843 3 "g) " }{XPPEDIT 844 1 "cos(3*x)-2*sin(3* x);" "6#,&-%$cosG6#*&\"\"$\"\"\"%\"xGF)F)*&\"\"#F)-%$sinG6#*&F(F)F*F)F )!\"\"" }{TEXT 845 9 " h) " }{XPPEDIT 846 1 "cos(6*x)-2*sin(6*x); " "6#,&-%$cosG6#*&\"\"'\"\"\"%\"xGF)F)*&\"\"#F)-%$sinG6#*&F(F)F*F)F)! \"\"" }{TEXT 847 12 " i) " }{XPPEDIT 848 1 "3*cos(36*x)+sin(36 *x);" "6#,&*&\"\"$\"\"\"-%$cosG6#*&\"#OF&%\"xGF&F&F&-%$sinG6#*&F+F&F,F &F&" }{TEXT 849 2 " " }}{PARA 257 "" 0 "" {TEXT 850 3 "j) " } {XPPEDIT 851 1 "2*cos(45*x)+sin(45*x);" "6#,&*&\"\"#\"\"\"-%$cosG6#*& \"#XF&%\"xGF&F&F&-%$sinG6#*&F+F&F,F&F&" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT 757 9 "Solution:" }{TEXT -1 2 " " }{TEXT 758 5 "( a )" }{TEXT 760 1 " " }{TEXT -1 0 "" }{TEXT 759 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "ds olve(9*diff(y(x),x$2)+36*y(x) = 0,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*&%$_C1G\"\"\"-%$cosG6#,$F'\"\"#F+F+*&%$_C2GF+-% $sinGF.F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 852 47 " 6. Which one of the \+ following expressions for " }{XPPEDIT 257 1 "y(x);" "6#-%\"yG6#%\"xG" }{TEXT 683 43 " is a solution to the differential equation" }{XPPEDIT 350 1 "diff(y(x),`$`(x,2))-4*diff(y(x),x)+5*y(x) = 0;" "6#/,(-%%diffG6 $-%\"yG6#%\"xG-%\"$G6$F+\"\"#\"\"\"*&\"\"%F0-F&6$-F)6#F+F+F0!\"\"*&\" \"&F0-F)6#F+F0F0\"\"!" }{TEXT 351 2 "? " }}{PARA 3 "" 0 "" {TEXT 362 6 "a) " }{XPPEDIT 352 0 "7*exp(x)+11*exp(2*x);" "6#,&*&\"\"(\"\"\"- %$expG6#%\"xGF&F&*&\"#6F&-F(6#*&\"\"#F&F*F&F&F&" }{TEXT 345 7 "\nb) \+ " }{XPPEDIT 353 0 "7*exp(x)+11*exp(-x);" "6#,&*&\"\"(\"\"\"-%$expG6#% \"xGF&F&*&\"#6F&-F(6#,$F*!\"\"F&F&" }{TEXT 346 7 "\nc) " }{XPPEDIT 354 0 "7*exp(4*x)+11*exp(5*x);" "6#,&*&\"\"(\"\"\"-%$expG6#*&\"\"%F&% \"xGF&F&F&*&\"#6F&-F(6#*&\"\"&F&F,F&F&F&" }{TEXT 347 7 "\nd) " } {XPPEDIT 355 0 "7*exp(2*x)*cos(x)+11*exp(2*x)*sin(x);" "6#,&*(\"\"(\" \"\"-%$expG6#*&\"\"#F&%\"xGF&F&-%$cosG6#F,F&F&*(\"#6F&-F(6#*&F+F&F,F&F &-%$sinG6#F,F&F&" }{TEXT 348 10 " \ne) " }{XPPEDIT 356 0 "7*exp(2 *x)*cos(2*x)+11*exp(2*x)*sin(2*x);" "6#,&*(\"\"(\"\"\"-%$expG6#*&\"\"# F&%\"xGF&F&-%$cosG6#*&F+F&F,F&F&F&*(\"#6F&-F(6#*&F+F&F,F&F&-%$sinG6#*& F+F&F,F&F&F&" }{TEXT 342 8 "\nf) " }{XPPEDIT 357 0 "7*exp(2*x)*cos (3*x)+11*exp(2*x)*sin(3*x);" "6#,&*(\"\"(\"\"\"-%$expG6#*&\"\"#F&%\"xG F&F&-%$cosG6#*&\"\"$F&F,F&F&F&*(\"#6F&-F(6#*&F+F&F,F&F&-%$sinG6#*&F1F& F,F&F&F&" }{TEXT 341 8 "\ng) " }{XPPEDIT 358 0 "7*exp(-2*x)*cos(x) +11*exp(-2*x)*sin(x);" "6#,&*(\"\"(\"\"\"-%$expG6#,$*&\"\"#F&%\"xGF&! \"\"F&-%$cosG6#F-F&F&*(\"#6F&-F(6#,$*&F,F&F-F&F.F&-%$sinG6#F-F&F&" } {TEXT 340 6 "\nh) " }{XPPEDIT 359 0 "7*exp(-2*x)*cos(2*x)+11*exp(-2* x)*sin(2*x);" "6#,&*(\"\"(\"\"\"-%$expG6#,$*&\"\"#F&%\"xGF&!\"\"F&-%$c osG6#*&F,F&F-F&F&F&*(\"#6F&-F(6#,$*&F,F&F-F&F.F&-%$sinG6#*&F,F&F-F&F&F &" }{TEXT 343 7 "\ni) " }{XPPEDIT 360 0 "7*exp(x)*cos(2*x)+11*exp(x )*sin(2*x);" "6#,&*(\"\"(\"\"\"-%$expG6#%\"xGF&-%$cosG6#*&\"\"#F&F*F&F &F&*(\"#6F&-F(6#F*F&-%$sinG6#*&F/F&F*F&F&F&" }{TEXT 344 7 "\nj) " } {XPPEDIT 361 0 "7*exp(-x)*cos(2*x)+11*exp(-x)*sin(2*x);" "6#,&*(\"\"( \"\"\"-%$expG6#,$%\"xG!\"\"F&-%$cosG6#*&\"\"#F&F+F&F&F&*(\"#6F&-F(6#,$ F+F,F&-%$sinG6#*&F1F&F+F&F&F&" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 753 9 "Solution:" }{TEXT -1 2 " " }{TEXT 754 5 "( d )" }{TEXT 756 1 " " }{TEXT -1 0 "" }{TEXT 755 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "dsolve( diff(y(x), x$2)-4*diff(y(x),x)+5*y(x) = 0,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*(%$_C1G\"\"\"-%$expG6#,$F'\"\"#F+-%$si nGF&F+F+*(%$_C2GF+F,F+-%$cosGF&F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 854 9 " 7. Let " }{XPPEDIT 364 1 "y( x) = Sum(c[n]*x^n,n = 0 .. infinity);" "6#/-%\"yG6#%\"xG-%$SumG6$*&&% \"cG6#%\"nG\"\"\")F'F/F0/F/;\"\"!%)infinityG" }{TEXT 365 78 " denote t he power series of the unique solution of the initial value problem \+ " }{XPPEDIT 366 1 "y*`'`(x) = x*y(x)+x^2,y(0) = 1;" "6$/*&%\"yG\"\"\"- %\"'G6#%\"xGF&,&*&F*F&-F%6#F*F&F&*$F*\"\"#F&/-F%6#\"\"!F&" }{TEXT 367 25 " . What is the value of " }{XPPEDIT 368 1 "c[4];" "6#&%\"cG6#\"\" %" }{TEXT 369 103 "? \n\na) 1 b) 1/2 c) 1/3 d) 2/3 e) 1/4 \+ f) 3/4 g) 1/8 h) 3/8 i) 1/12 j) 5/12" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 761 9 "Solution:" }{TEXT -1 2 " " }{TEXT 762 5 "( g )" }{TEXT 764 1 " " }{TEXT -1 0 "" }{TEXT 763 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "dsolve(\{diff(y(x),x) = x*y(x)+x^2, y(0) = 1\},y(x), \+ series);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG+/F'\"\"\"\" \"!#F)\"\"#F,#F)\"\"$F.#F)\"\")\"\"%#F)\"#:\"\"&-%\"OG6#F)\"\"'" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 380 8 "8. Let " }{XPPEDIT 381 1 "y(x) = Sum(c[n]*x^n,n = 0 .. infinity);" "6#/-%\" yG6#%\"xG-%$SumG6$*&&%\"cG6#%\"nG\"\"\")F'F/F0/F/;\"\"!%)infinityG" } {TEXT 382 67 " denote a power series of a solution of the differential equation " }{XPPEDIT 383 1 "y*`''`(x)+(x-1)*y(x) = 0;" "6#/,&*&%\"yG \"\"\"-%#''G6#%\"xGF'F'*&,&F+F'F'!\"\"F'-F&6#F+F'F'\"\"!" }{TEXT 384 73 " . Then the coefficients satisfy which of the following recurrenc es for " }{XPPEDIT 501 0 "1 <= n;" "6#1\"\"\"%\"nG" }{TEXT 500 2 " ?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "a) " } {XPPEDIT 256 0 "c[n+1] = c[n]/(n+1);" "6#/&%\"cG6#,&%\"nG\"\"\"F)F)*&& F%6#F(F),&F(F)F)F)!\"\"" }{TEXT -1 26 " b) " } {XPPEDIT 258 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 -1 8 "\n\nc) " } {XPPEDIT 259 0 "c[n+1] = -c[n]/(n+1);" "6#/&%\"cG6#,&%\"nG\"\"\"F)F),$ *&&F%6#F(F),&F(F)F)F)!\"\"F/" }{TEXT -1 21 " d) " } {XPPEDIT 260 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 -1 7 "\n\ne) " } {XPPEDIT 261 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 -1 19 " f) \+ " }{XPPEDIT 262 0 "c[n+2] = (c[n]+c[n-1])/((n+1)*(n+2));" "6#/&%\"cG6# ,&%\"nG\"\"\"\"\"#F)*&,&&F%6#F(F)&F%6#,&F(F)F)!\"\"F)F)*&,&F(F)F)F)F), &F(F)F*F)F)F2" }{TEXT -1 6 "\n\ng) " }{XPPEDIT 263 0 "c[n+2] = (c[n]- c[n-1])/(n+1);" "6#/&%\"cG6#,&%\"nG\"\"\"\"\"#F)*&,&&F%6#F(F)&F%6#,&F( F)F)!\"\"F2F),&F(F)F)F)F2" }{TEXT -1 18 " " }{TEXT 385 5 "h) " }{XPPEDIT 264 0 "c[n+2] = (c[n]-c[n-1])/(n+2);" "6#/&%\" cG6#,&%\"nG\"\"\"\"\"#F)*&,&&F%6#F(F)&F%6#,&F(F)F)!\"\"F2F),&F(F)F*F)F 2" }{TEXT 378 7 "\n\ni) " }{XPPEDIT 265 0 "c[n+2] = (c[n]+c[n-1])/(n +2);" "6#/&%\"cG6#,&%\"nG\"\"\"\"\"#F)*&,&&F%6#F(F)&F%6#,&F(F)F)!\"\"F )F),&F(F)F*F)F2" }{TEXT 379 22 " j) " }{XPPEDIT 267 0 "c[n+2] = (c[n]-c[n-1])/((n+1)*(n+2));" "6#/&%\"cG6#,&%\"nG\"\"\"\" \"#F)*&,&&F%6#F(F)&F%6#,&F(F)F)!\"\"F2F)*&,&F(F)F)F)F),&F(F)F*F)F)F2" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 797 9 "Solution:" }{TEXT -1 2 " " }{TEXT 798 5 "( j )" }{TEXT 800 1 " " }{TEXT -1 0 "" }{TEXT 799 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(Slode):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "FPseries(diff(y(x),x$2)+(x-1)*y(x) = 0, y(x), c(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%*FPSstructG6$,*&%#_CG6#\"\"!\"\"\"*&&F(6#F +F+%\"xGF+F+*(#F+\"\"#F+F'F+)F/F2F+F+-%$SumG6$*&-%\"cG6#%\"nGF+)F/F;F+ /F;;\"\"$%)infinityGF+,(*&,&*$)F;F2F+F+F;!\"\"F+F8F+F+-F96#,&!\"#F+F;F +FF-F96#,&F;F+F?FFF+-%&TABLEG6#7)/%$P_nGF " 0 "" {MPLTEXT 1 0 42 "subs(n=n+2,(n^2-n)*c(n)-c(-2 +n)+c(n-3)=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&,(*$),&%\"nG\" \"\"\"\"#F+F,F+F+F*!\"\"F,F-F+-%\"cG6#F)F+F+-F/6#F*F--F/6#,&F-F+F*F+F+ \"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "expand(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/,,-%\"cG6#%\"nG!\"\"-F&6#,&F)\"\"\"F( F-F-*&-F&6#,&F(F-\"\"#F-F-)F(F2F-F-*(\"\"$F-F/F-F(F-F-*&F2F-F/F-F-\"\" !" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "isolate(%,c(n+2));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"cG6#,&%\"nG\"\"\"\"\"#F)*&,&-F%6# F(F)-F%6#,&!\"\"F)F(F)F2F),(*$)F(F*F)F)*&\"\"$F)F(F)F)F*F)F2" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 370 155 " 9. When we seek a Frobenius series centered at 0 for a solution, which of the following equations is the indicial equation of the diff erential equation " }{XPPEDIT 396 0 "x*y*`''`(x)+y*`'`(x)/2-y(x) = 0 ;" "6#/,(*(%\"xG\"\"\"%\"yGF'-%#''G6#F&F'F'*(F(F'-%\"'G6#F&F'\"\"#!\" \"F'-F(6#F&F1\"\"!" }{TEXT 397 5 "?\na) " }{XPPEDIT 256 1 "r^2-r/2 = 0 ;" "6#/,&*$%\"rG\"\"#\"\"\"*&F&F(F'!\"\"F*\"\"!" }{TEXT 386 6 " \+ " }}{PARA 3 "" 0 "" {TEXT 698 4 "b) " }{XPPEDIT 258 1 "r^2-r/2-1 = 0; " "6#/,(*$%\"rG\"\"#\"\"\"*&F&F(F'!\"\"F*F(F*\"\"!" }{TEXT 387 6 " \+ " }}{PARA 3 "" 0 "" {TEXT 699 4 "c) " }{XPPEDIT 259 1 "r^2+r/2 = 0; " "6#/,&*$%\"rG\"\"#\"\"\"*&F&F(F'!\"\"F(\"\"!" }{TEXT 388 6 " " }}{PARA 3 "" 0 "" {TEXT 700 4 "d) " }{XPPEDIT 260 1 "r^2+r/2-1 = 0;" "6#/,(*$%\"rG\"\"#\"\"\"*&F&F(F'!\"\"F(F(F*\"\"!" }{TEXT 389 6 " \+ " }}{PARA 3 "" 0 "" {TEXT 701 4 "e) " }{XPPEDIT 261 1 "r/2-1 = 0;" "6 #/,&*&%\"rG\"\"\"\"\"#!\"\"F'F'F)\"\"!" }{TEXT 390 6 "\n\nf) " } {XPPEDIT 262 1 "x*r^2+r/2-1 = 0;" "6#/,(*&%\"xG\"\"\"*$%\"rG\"\"#F'F'* &F)F'F*!\"\"F'F'F,\"\"!" }{TEXT 391 9 " " }}{PARA 3 "" 0 "" {TEXT 702 4 "g) " }{XPPEDIT 266 0 "r^2-r/2+1 = 0;" "6#/,(*$%\"rG\"\"# \"\"\"*&F&F(F'!\"\"F*F(F(\"\"!" }{TEXT 394 9 " " }}{PARA 3 "" 0 "" {TEXT 703 3 "h) " }{XPPEDIT 263 1 "r^2+r/2-x*r = 0;" "6#/,(*$%\"r G\"\"#\"\"\"*&F&F(F'!\"\"F(*&%\"xGF(F&F(F*\"\"!" }{TEXT 392 9 " \+ " }}{PARA 3 "" 0 "" {TEXT 704 3 "i) " }{XPPEDIT 264 1 "-r/2-1 = 0;" "6#/,&*&%\"rG\"\"\"\"\"#!\"\"F)F'F)\"\"!" }{TEXT 393 93 "\n\nj) There \+ is no indicial equation because 0 is not a regular singular point of t he equation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 793 9 "Solution:" }{TEXT -1 2 " " } {TEXT 794 5 "( a )" }{TEXT 796 1 " " }{TEXT -1 0 "" }{TEXT 795 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "x*(x*diff(y(x),x) + diff(y(x ),x)/2-y(x) = 0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%\"xG\"\"\",(* &F%F&-%%diffG6$-%\"yG6#F%F%F&F&*&#F&\"\"#F&F)F&F&F,!\"\"F&\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "expand(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&)%\"xG\"\"#\"\"\"-%%diffG6$-%\"yG6#F'F'F)F)*(#F )F(F)F'F)F*F)F)*&F'F)F-F)!\"\"\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "r*(r-1)+1/2*r-0=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/,&*&%\"rG\"\"\",&F&F'F'!\"\"F'F'*&#F'\"\"#F'F&F'F'\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "expand(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*$)%\"rG\"\"#\"\"\"F)*&#F)F(F)F'F)!\"\"\"\"!" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 405 96 "10.Which one of the following Frobenius series might be a solution of the differential equation " }{XPPEDIT 407 1 "9*x^2*diff(y(x),`$`(x,2) )+3*x*diff(y(x),x)+y(x) = 0;" "6#/,(*(\"\"*\"\"\"*$%\"xG\"\"#F'-%%diff G6$-%\"yG6#F)-%\"$G6$F)F*F'F'*(\"\"$F'F)F'-F,6$-F/6#F)F)F'F'-F/6#F)F' \"\"!" }{TEXT 406 4 " ? \n" }}{PARA 3 "" 0 "" {TEXT 409 4 "a) " } {XPPEDIT 426 1 "y(x) = Sum(c[n]*x^(n-1/3),n = 0 .. infinity);" "6#/-% \"yG6#%\"xG-%$SumG6$*&&%\"cG6#%\"nG\"\"\")F',&F/F0*&F0F0\"\"$!\"\"F5F0 /F/;\"\"!%)infinityG" }{TEXT 410 11 " b) " }{XPPEDIT 427 1 "y(x ) = Sum(c[n]*x^(n+1/3),n = 0 .. infinity);" "6#/-%\"yG6#%\"xG-%$SumG6$ *&&%\"cG6#%\"nG\"\"\")F',&F/F0*&F0F0\"\"$!\"\"F0F0/F/;\"\"!%)infinityG " }{TEXT 411 1 " " }{TEXT 403 1 " " }{TEXT 404 4 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 416 0 "" }{TEXT 408 4 "c) \+ " }{XPPEDIT 417 1 "y(x) = Sum(c[n]*x^(n+2/3),n = 0 .. infinity);" "6#/ -%\"yG6#%\"xG-%$SumG6$*&&%\"cG6#%\"nG\"\"\")F',&F/F0*&\"\"#F0\"\"$!\" \"F0F0/F/;\"\"!%)infinityG" }{TEXT 412 15 " d) " }{XPPEDIT 418 1 "y(x) = Sum(c[n]*x^(n-2/3),n = 0 .. infinity);" "6#/-%\"yG6#%\"x G-%$SumG6$*&&%\"cG6#%\"nG\"\"\")F',&F/F0*&\"\"#F0\"\"$!\"\"F6F0/F/;\" \"!%)infinityG" }{TEXT 398 7 " " }}{PARA 0 "" 0 "" {TEXT 413 3 " e) " }{XPPEDIT 419 1 "y(x) = Sum(c[n]*x^(n+1/4),n = 0 .. infinity);" " 6#/-%\"yG6#%\"xG-%$SumG6$*&&%\"cG6#%\"nG\"\"\")F',&F/F0*&F0F0\"\"%!\" \"F0F0/F/;\"\"!%)infinityG" }{TEXT 399 16 " f) " } {XPPEDIT 420 1 "y(x) = Sum(c[n]*x^(n-1/4),n = 0 .. infinity);" "6#/-% \"yG6#%\"xG-%$SumG6$*&&%\"cG6#%\"nG\"\"\")F',&F/F0*&F0F0\"\"%!\"\"F5F0 /F/;\"\"!%)infinityG" }{TEXT 415 6 " \n\ng) " }{XPPEDIT 421 1 "y(x) = \+ Sum(c[n]*x^(n-1/9),n = 0 .. infinity);" "6#/-%\"yG6#%\"xG-%$SumG6$*&&% \"cG6#%\"nG\"\"\")F',&F/F0*&F0F0\"\"*!\"\"F5F0/F/;\"\"!%)infinityG" } {TEXT 400 15 " h) " }{XPPEDIT 422 1 "y(x) = Sum(c[n]*x^(n+1 /9),n = 0 .. infinity);" "6#/-%\"yG6#%\"xG-%$SumG6$*&&%\"cG6#%\"nG\"\" \")F',&F/F0*&F0F0\"\"*!\"\"F0F0/F/;\"\"!%)infinityG" }{TEXT 401 5 " \+ " }}{PARA 0 "" 0 "" {TEXT 414 4 "i) " }{XPPEDIT 423 1 "y(x) = Sum(c [n]*x^(n+3/2),n = 0 .. infinity);" "6#/-%\"yG6#%\"xG-%$SumG6$*&&%\"cG6 #%\"nG\"\"\")F',&F/F0*&\"\"$F0\"\"#!\"\"F0F0/F/;\"\"!%)infinityG" } {TEXT 402 16 " j) " }{XPPEDIT 424 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" }}{PARA 0 " " 0 "" {TEXT 425 0 "" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 789 9 "So lution:" }{TEXT -1 2 " " }{TEXT 790 5 "( b )" }{TEXT 792 1 " " } {TEXT -1 0 "" }{TEXT 791 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "ode := 9*x^2*diff(y(x),x$2)+ 3*x*diff(y(x),x)+y(x) = 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$odeG/ ,(*&)%\"xG\"\"#\"\"\"-%%diffG6$-%\"yG6#F)-%\"$G6$F)F*F+\"\"**(\"\"$F+F )F+-F-6$F/F)F+F+F/F+\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "ode/9;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&)%\"xG\"\"#\"\"\"-%%di ffG6$-%\"yG6#F'-%\"$G6$F'F(F)F)*(#F)\"\"$F)F'F)-F+6$F-F'F)F)*&#F)\"\"* F)F-F)F)\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "solve(r*(r -1)+r/3+1/9,r);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$#\"\"\"\"\"$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 587 59 "11. W hich expression is the inverse Laplace transform of " }{XPPEDIT 588 1 "3*s/(s^2+8*s+25);" "6#*(\"\"$\"\"\"%\"sGF%,(*$F&\"\"#F%*&\"\")F%F&F %F%\"#DF%!\"\"" }{TEXT 589 2 " ?" }}{PARA 0 "" 0 "" {TEXT 590 0 "" }} {PARA 0 "" 0 "" {TEXT 591 3 "a) " }{XPPEDIT 592 0 "4*exp(-4*t)*cos(5*t )+3*exp(-4*t)*sin(5*t);" "6#,&*(\"\"%\"\"\"-%$expG6#,$*&F%F&%\"tGF&!\" \"F&-%$cosG6#*&\"\"&F&F,F&F&F&*(\"\"$F&-F(6#,$*&F%F&F,F&F-F&-%$sinG6#* &F2F&F,F&F&F&" }{TEXT 593 8 " b) " }{XPPEDIT 594 0 "3*exp(-4*t)*co s(4*t)+4*exp(-4*t)*sin(4*t);" "6#,&*(\"\"$\"\"\"-%$expG6#,$*&\"\"%F&% \"tGF&!\"\"F&-%$cosG6#*&F,F&F-F&F&F&*(F,F&-F(6#,$*&F,F&F-F&F.F&-%$sinG 6#*&F,F&F-F&F&F&" }{TEXT 595 7 " \nc) " }{XPPEDIT 596 1 "3*exp(-4*t) *cos(3*t)-4*exp(-4*t)*sin(3*t);" "6#,&*(\"\"$\"\"\"-%$expG6#,$*&\"\"%F &%\"tGF&!\"\"F&-%$cosG6#*&F%F&F-F&F&F&*(F,F&-F(6#,$*&F,F&F-F&F.F&-%$si nG6#*&F%F&F-F&F&F." }{TEXT 597 8 " d) " }{XPPEDIT 598 0 "5*exp(-4* t)*cos(3*t)-3*exp(-4*t)*sin(3*t);" "6#,&*(\"\"&\"\"\"-%$expG6#,$*&\"\" %F&%\"tGF&!\"\"F&-%$cosG6#*&\"\"$F&F-F&F&F&*(F3F&-F(6#,$*&F,F&F-F&F.F& -%$sinG6#*&F3F&F-F&F&F." }{TEXT 599 1 " " }}{PARA 0 "" 0 "" {TEXT 600 3 "e) " }{XPPEDIT 601 0 "5*exp(-2*t)*cos(3*t)-3*exp(-2*t)*sin(3*t);" " 6#,&*(\"\"&\"\"\"-%$expG6#,$*&\"\"#F&%\"tGF&!\"\"F&-%$cosG6#*&\"\"$F&F -F&F&F&*(F3F&-F(6#,$*&F,F&F-F&F.F&-%$sinG6#*&F3F&F-F&F&F." }{TEXT 602 9 " f) " }{XPPEDIT 603 0 "5*exp(-t)*cos(5*t)+4*exp(-t)*sin(5*t); " "6#,&*(\"\"&\"\"\"-%$expG6#,$%\"tG!\"\"F&-%$cosG6#*&F%F&F+F&F&F&*(\" \"%F&-F(6#,$F+F,F&-%$sinG6#*&F%F&F+F&F&F&" }{TEXT 604 2 " " }}{PARA 0 "" 0 "" {TEXT 605 4 "g) " }{XPPEDIT 606 0 "3*exp(-2*t)*cos(5*t);" " 6#*(\"\"$\"\"\"-%$expG6#,$*&\"\"#F%%\"tGF%!\"\"F%-%$cosG6#*&\"\"&F%F,F %F%" }{TEXT 607 43 " h) " } {XPPEDIT 608 0 "3/8*exp(-2*t)*cos(5*t);" "6#**\"\"$\"\"\"\"\")!\"\"-%$ expG6#,$*&\"\"#F%%\"tGF%F'F%-%$cosG6#*&\"\"&F%F.F%F%" }{TEXT 609 3 " \+ " }}{PARA 0 "" 0 "" {TEXT 610 3 "i) " }{XPPEDIT 611 0 "3/8*exp(-4*t)* cos(5*t);" "6#**\"\"$\"\"\"\"\")!\"\"-%$expG6#,$*&\"\"%F%%\"tGF%F'F%-% $cosG6#*&\"\"&F%F.F%F%" }{TEXT 612 44 " \+ j) " }{XPPEDIT 256 0 "3/8*exp(-4*t)*sin(5*t);" "6#**\"\"$\" \"\"\"\")!\"\"-%$expG6#,$*&\"\"%F%%\"tGF%F'F%-%$sinG6#*&\"\"&F%F.F%F% " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 765 9 "Solut ion:" }{TEXT -1 2 " " }{TEXT 766 5 "( c )" }{TEXT 768 1 " " }{TEXT -1 0 "" }{TEXT 767 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "with(inttrans): invlaplace(3*s/(s^2 +8*s+25),s,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&-%$expG6#,$%\"tG !\"%\"\"\"-%$cosG6#,$F)\"\"$F+F0*(\"\"%F+F%F+-%$sinGF.F+!\"\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 0 {PARA 3 "" 0 "" {TEXT 615 60 "12. Which expression is the inve rse Laplace transform of " }{XPPEDIT 257 1 "exp(-s)/(s+2);" "6#*&-% $expG6#,$%\"sG!\"\"\"\"\",&F(F*\"\"#F*F)" }{TEXT 613 34 " ? \n(In the \+ answers, the function " }{XPPEDIT 616 1 "u;" "6#%\"uG" }{TEXT 614 110 " is the Heaviside function: \n \+ u(t) = 1 for t > 0 and 0 for t < 0.)" }}{PARA 0 "" 0 "" {TEXT 617 0 "" }}{PARA 0 "" 0 "" {TEXT 618 3 "a) " }{XPPEDIT 619 1 "u(t-1)*e xp(-2*t);" "6#*&-%\"uG6#,&%\"tG\"\"\"F)!\"\"F)-%$expG6#,$*&\"\"#F)F(F) F*F)" }{TEXT 620 18 " b) " }{XPPEDIT 621 1 "u(t-1)*exp(- 2*t-2);" "6#*&-%\"uG6#,&%\"tG\"\"\"F)!\"\"F)-%$expG6#,&*&\"\"#F)F(F)F* F0F*F)" }{TEXT 622 2 " " }}{PARA 0 "" 0 "" {TEXT 623 3 "c) " } {XPPEDIT 624 1 "u(t)*exp(-2*t+2);" "6#*&-%\"uG6#%\"tG\"\"\"-%$expG6#,& *&\"\"#F(F'F(!\"\"F.F(F(" }{TEXT 625 18 " d) " } {XPPEDIT 626 1 "u(t-1)*exp(-2*t+2);" "6#*&-%\"uG6#,&%\"tG\"\"\"F)!\"\" F)-%$expG6#,&*&\"\"#F)F(F)F*F0F)F)" }{TEXT 627 3 " " }}{PARA 0 "" 0 "" {TEXT 628 3 "e) " }{XPPEDIT 629 1 "u(t+1)*exp(-2*t-2);" "6#*&-%\"uG 6#,&%\"tG\"\"\"F)F)F)-%$expG6#,&*&\"\"#F)F(F)!\"\"F/F0F)" }{TEXT 630 12 " f) " }{XPPEDIT 631 1 "u(t-2)*exp(-t-2);" "6#*&-%\"uG6#,&% \"tG\"\"\"\"\"#!\"\"F)-%$expG6#,&F(F+F*F+F)" }{TEXT 632 3 " " }} {PARA 0 "" 0 "" {TEXT 633 3 "g) " }{XPPEDIT 634 1 "u(t-2)*exp(-t+2);" "6#*&-%\"uG6#,&%\"tG\"\"\"\"\"#!\"\"F)-%$expG6#,&F(F+F*F)F)" }{TEXT 635 15 " h) " }{XPPEDIT 636 1 "u(t+2)*exp(-t-2);" "6#*&-%\" uG6#,&%\"tG\"\"\"\"\"#F)F)-%$expG6#,&F(!\"\"F*F/F)" }{TEXT 637 2 " " }}{PARA 0 "" 0 "" {TEXT 638 3 "i) " }{XPPEDIT 639 1 "u(t-1)*exp(-t-1); " "6#*&-%\"uG6#,&%\"tG\"\"\"F)!\"\"F)-%$expG6#,&F(F*F)F*F)" }{TEXT 640 16 " j) " }{XPPEDIT 690 1 "u(t-1)*exp(-t+1);" "6#*&-% \"uG6#,&%\"tG\"\"\"F)!\"\"F)-%$expG6#,&F(F*F)F)F)" }{TEXT 689 5 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 769 9 "Solut ion:" }{TEXT -1 2 " " }{TEXT 770 5 "( d )" }{TEXT 772 1 " " }{TEXT -1 0 "" }{TEXT 771 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "with(inttrans): invlaplace(exp(-s)/ (s+2),s,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%*HeavisideG6#,&%\"t G\"\"\"F)!\"\"F)-%$expG6#,&F(!\"#\"\"#F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 641 8 "13. If " }{XPPEDIT 642 0 "vector([a, b, c, d]);" "6#-%'v ectorG6#7&%\"aG%\"bG%\"cG%\"dG" }{TEXT 643 24 " is an eigenvector of \+ " }{XPPEDIT 644 1 "matrix([[3, -1, 2, -2], [1, 1, 1, 0], [0, 0, 2, 1] , [0, 0, 0, 2]]);" "6#-%'matrixG6#7&7&\"\"$,$\"\"\"!\"\"\"\"#,$F,F+7&F *F*F*\"\"!7&F/F/F,F*7&F/F/F/F," }{TEXT 645 19 " then what is a/b?" }} {PARA 0 "" 0 "" {TEXT 646 66 "a) undefined since b = 0 b) 1 \+ c) 2 " }}{PARA 0 "" 0 "" {TEXT 705 67 "d) 1/2 \+ e) 3 f) -1 " }}{PARA 0 " " 0 "" {TEXT 706 69 "g) -2 h) -1 /2 i) - 3 " }}{PARA 0 "" 0 "" {TEXT 707 27 "j) not uniquely determined" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 773 9 "Solution:" }{TEXT -1 2 " " }{TEXT 774 5 "( b )" }{TEXT 776 1 " " }{TEXT -1 0 "" }{TEXT 775 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "with(linalg):\nA := matrix([ [3, -1, 2, -2], [1, 1, 1, 0], [0, 0, 2, 1], [0, 0, 0, 2]]);" }}{PARA 7 "" 1 "" {TEXT 371 45 "Warning, the name hilbert has been redefined\n " }}{PARA 7 "" 1 "" {TEXT 372 80 "Warning, the protected names norm an d trace have been redefined and unprotected\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7&7&\"\"$!\"\"\"\"#!\"#7&\"\"\"F/F/\" \"!7&F0F0F,F/7&F0F0F0F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 " eigenvects(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\"\"#\"\"%<#-%'vec torG6#7&\"\"\"F+\"\"!F," }}}{PARA 3 "" 0 "" {TEXT 373 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 647 29 "14. The matrix of the system" }}{PARA 0 "" 0 "" {TEXT 648 2 " " }}{PARA 0 " " 0 "" {TEXT 649 6 " " }{XPPEDIT 650 1 "diff(x(t),t) = x-3*y;" "6 #/-%%diffG6$-%\"xG6#%\"tGF*,&F(\"\"\"*&\"\"$F,%\"yGF,!\"\"" }}{PARA 0 "" 0 "" {TEXT 651 5 " " }{XPPEDIT 652 1 "diff(y(t),t) = 2*x-4*y;" "6#/-%%diffG6$-%\"yG6#%\"tGF*,&*&\"\"#\"\"\"%\"xGF.F.*&\"\"%F.F(F.!\" \"" }}{PARA 0 "" 0 "" {TEXT 653 0 "" }}{PARA 0 "" 0 "" {TEXT 654 38 "h as eigenvectors [1,1] and [3,2]. If " }{XPPEDIT 655 1 "x(0) = 1;" "6# /-%\"xG6#\"\"!\"\"\"" }{TEXT 656 6 " and " }{XPPEDIT 657 1 "y(0) = 2; " "6#/-%\"yG6#\"\"!\"\"#" }{TEXT 658 14 " then what is " }{XPPEDIT 659 1 "x(t);" "6#-%\"xG6#%\"tG" }{TEXT 660 2 " ?" }}{PARA 0 "" 0 "" {TEXT 661 0 "" }}{PARA 0 "" 0 "" {TEXT 662 4 "a) " }{XPPEDIT 663 0 "4 *exp(-2*t)-3*exp(-t);" "6#,&*&\"\"%\"\"\"-%$expG6#,$*&\"\"#F&%\"tGF&! \"\"F&F&*&\"\"$F&-F(6#,$F-F.F&F." }{TEXT 664 14 " b) " } {XPPEDIT 665 0 "2*exp(2*t)-5*exp(-t);" "6#,&*&\"\"#\"\"\"-%$expG6#*&F% F&%\"tGF&F&F&*&\"\"&F&-F(6#,$F+!\"\"F&F1" }}{PARA 0 "" 0 "" {TEXT 666 5 "c) " }{XPPEDIT 667 0 "2*exp(-2*t)-5*exp(-t);" "6#,&*&\"\"#\"\"\"- %$expG6#,$*&F%F&%\"tGF&!\"\"F&F&*&\"\"&F&-F(6#,$F,F-F&F-" }{TEXT 668 12 " d) " }{XPPEDIT 669 0 "2*exp(2*t)+exp(-t);" "6#,&*&\"\"#\" \"\"-%$expG6#*&F%F&%\"tGF&F&F&-F(6#,$F+!\"\"F&" }}{PARA 0 "" 0 "" {TEXT 670 3 "e) " }{XPPEDIT 671 0 "2*t*exp(2*t)-5*exp(-t);" "6#,&*(\" \"#\"\"\"%\"tGF&-%$expG6#*&F%F&F'F&F&F&*&\"\"&F&-F)6#,$F'!\"\"F&F1" } {TEXT 672 15 " f) " }{XPPEDIT 673 0 "2*exp(2*t)+(-3*exp(t)) ;" "6#,&*&\"\"#\"\"\"-%$expG6#*&F%F&%\"tGF&F&F&,$*&\"\"$F&-F(6#F+F&!\" \"F&" }}{PARA 0 "" 0 "" {TEXT 674 3 "g) " }{XPPEDIT 675 0 "exp(-2*t)-6 *exp(t);" "6#,&-%$expG6#,$*&\"\"#\"\"\"%\"tGF*!\"\"F**&\"\"'F*-F%6#F+F *F," }{TEXT 676 23 " h) " }{XPPEDIT 677 0 "3*exp(-2 *t)+2*exp(-t);" "6#,&*&\"\"$\"\"\"-%$expG6#,$*&\"\"#F&%\"tGF&!\"\"F&F& *&F,F&-F(6#,$F-F.F&F&" }{TEXT 678 2 " " }}{PARA 0 "" 0 "" {TEXT 679 4 "i) " }{XPPEDIT 680 0 "exp(-2*t)-4*exp(3*t);" "6#,&-%$expG6#,$*&\" \"#\"\"\"%\"tGF*!\"\"F**&\"\"%F*-F%6#*&\"\"$F*F+F*F*F," }{TEXT 681 16 " j) " }{XPPEDIT 682 0 "exp(2*t)-4*exp(t);" "6#,&-%$expG6# *&\"\"#\"\"\"%\"tGF)F)*&\"\"%F)-F%6#F*F)!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 777 9 "Solution:" }{TEXT -1 2 " \+ " }{TEXT 778 5 "( a )" }{TEXT 780 1 " " }{TEXT -1 0 "" }{TEXT 779 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "dsolve(\{diff(x(t),t) = x(t)-3*y(t),diff(y(t),t) = 2*x(t)-4*y( t),x(0) = 1,y(0) = 2\},\{x(t),y(t)\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/-%\"yG6#%\"tG,&-%$expG6#,$F(!\"#\"\"%*&\"\"#\"\"\"-F+6#,$F(! \"\"F2F6/-%\"xGF',&F*F/*&\"\"$F2F3F2F6" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 580 16 "15. The system " }}{PARA 0 "" 0 "" {TEXT 573 5 " " }{XPPEDIT 581 1 "diff(x(t),t) = 14*x-2*x ^2-x*y;" "6#/-%%diffG6$-%\"xG6#%\"tGF*,(*&\"#9\"\"\"F(F.F.*&\"\"#F.*$F (F0F.!\"\"*&F(F.%\"yGF.F2" }{TEXT 574 2 " " }}{PARA 0 "" 0 "" {TEXT 575 5 " " }{XPPEDIT 582 1 "diff(y(t),t) = 16*y-2*y^2-x*y;" "6#/-%% diffG6$-%\"yG6#%\"tGF*,(*&\"#;\"\"\"F(F.F.*&\"\"#F.*$F(F0F.!\"\"*&%\"x GF.F(F.F2" }{TEXT 576 2 " " }}{PARA 3 "" 0 "" {TEXT 577 42 "has exact ly one critical point (a,b) with " }{XPPEDIT 583 1 "a*b <> 0;" "6#0*&% \"aG\"\"\"%\"bGF&\"\"!" }{TEXT 578 11 ". What is " }{XPPEDIT 584 1 "a *b;" "6#*&%\"aG\"\"\"%\"bGF%" }{TEXT 579 2 " ?" }}{PARA 0 "" 0 "" {TEXT 585 0 "" }}{PARA 0 "" 0 "" {TEXT 586 96 "a) 2 b) 4 c) 6 d) 8 e) 12 f) 16 g) 20 h) 24 i) 28 j) 32" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 781 9 "So lution:" }{TEXT -1 2 " " }{TEXT 782 5 "( h )" }{TEXT 784 1 " " } {TEXT -1 0 "" }{TEXT 783 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "solve(\{14*x-2*x^2-x*y=0,16* y-2*y^2-x*y=0\},\{x,y\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&<$/%\"yG \"\"!/%\"xGF&<$/F(\"\"(F$<$/F%\"\")F'<$/F(\"\"%/F%\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "subs(\{x = 4, y = 6\},\{14*x-2*x^2- x*y=0,16*y-2*y^2-x*y=0\}); # Verify" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #<#/\"\"!F%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 549 95 "16. The system of problem 15 also has critical point \+ (0,8). If we make the change of variables " }{XPPEDIT 550 1 "x = u,y = v+8;" "6$/%\"xG%\"uG/%\"yG,&%\"vG\"\"\"\"\")F*" }{TEXT 551 89 ", then what is the coefficient matrix M of the corresponding linear system i n u' and v' ?" }}{PARA 0 "" 0 "" {TEXT 552 3 "a) " }{XPPEDIT 553 1 "ma trix([[6, 0], [-8, -16]]);" "6#-%'matrixG6#7$7$\"\"'\"\"!7$,$\"\")!\" \",$\"#;F-" }{TEXT 554 8 " b) " }{XPPEDIT 555 1 "matrix([[6, 0], [ -16, 12]]);" "6#-%'matrixG6#7$7$\"\"'\"\"!7$,$\"#;!\"\"\"#7" }{TEXT 556 9 " c) " }{XPPEDIT 557 1 "matrix([[6, 0], [-16, 14]]);" "6#-% 'matrixG6#7$7$\"\"'\"\"!7$,$\"#;!\"\"\"#9" }{TEXT 558 8 " d) " } {XPPEDIT 559 1 "matrix([[14, -2], [16, -2]]);" "6#-%'matrixG6#7$7$\"#9 ,$\"\"#!\"\"7$\"#;,$F*F+" }{TEXT 560 7 " e) " }{XPPEDIT 561 1 "matr ix([[12, 0], [-14, 8]]);" "6#-%'matrixG6#7$7$\"#7\"\"!7$,$\"#9!\"\"\" \")" }{TEXT 562 2 " \n" }}{PARA 0 "" 0 "" {TEXT 563 4 "f) " } {XPPEDIT 564 1 "matrix([[12, 0], [14, 8]]);" "6#-%'matrixG6#7$7$\"#7\" \"!7$\"#9\"\")" }{TEXT 565 12 " g) " }{XPPEDIT 566 1 "matrix([ [16, 0], [-16, 8]]);" "6#-%'matrixG6#7$7$\"#;\"\"!7$,$F(!\"\"\"\")" } {TEXT 567 10 " h) " }{XPPEDIT 568 1 "matrix([[0, 6], [16, -12]]) ;" "6#-%'matrixG6#7$7$\"\"!\"\"'7$\"#;,$\"#7!\"\"" }{TEXT 569 9 " \+ i) " }{XPPEDIT 570 1 "matrix([[6, 4], [8, 16]]);" "6#-%'matrixG6#7$7$ \"\"'\"\"%7$\"\")\"#;" }{TEXT 571 9 " j) " }{XPPEDIT 572 1 "matri x([[0, 6], [12, 14]]);" "6#-%'matrixG6#7$7$\"\"!\"\"'7$\"#7\"#9" }} {PARA 0 "" 0 "" {TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 785 9 "Solution:" }{TEXT -1 2 " " }{TEXT 786 5 "( a )" }{TEXT 788 1 " " }{TEXT -1 0 "" }{TEXT 787 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Diff(u( t),t) = 14*u-2*u^2-u*(v+8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%Dif fG6$-%\"uG6#%\"tGF*,(F(\"#9*&\"\"#\"\"\")F(F.F/!\"\"*&F(F/,&%\"vGF/\" \")F/F/F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "expand(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%\"uG6#%\"tGF*,(F(\"\"'*& \"\"#\"\"\")F(F.F/!\"\"*&F(F/%\"vGF/F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "lhs(%) = mtaylor(rhs(%),[u,v],2);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%%DiffG6$-%\"uG6#%\"tGF*,$F(\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "Diff(v(t),t) = 16*(v+8)-2*(v+8)^2-u*(v+8) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%\"vG6#%\"tGF*,*F(\"# ;\"$G\"\"\"\"*&\"\"#F.),&F(F.\"\")F.F0F.!\"\"*&%\"uGF.F2F.F4" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "expand(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%\"vG6#%\"tGF*,*F(!#;*&\"\"#\"\"\")F(F.F /!\"\"*&%\"uGF/F(F/F1*&\"\")F/F3F/F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "lhs(%) = mtaylor(rhs(%),[u,v],2);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%%DiffG6$-%\"vG6#%\"tGF*,&F(!#;*&\"\")\"\"\"%\"uGF/ !\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 537 52 "17. Classify the critical point (0,8) of the system " }}{PARA 0 "" 0 "" {TEXT 692 5 " " }{XPPEDIT 261 1 "diff(x(t),t) = 14*x-2*x ^2-x*y;" "6#/-%%diffG6$-%\"xG6#%\"tGF*,(*&\"#9\"\"\"F(F.F.*&\"\"#F.*$F (F0F.!\"\"*&F(F.%\"yGF.F2" }{TEXT 693 2 " " }}{PARA 0 "" 0 "" {TEXT 694 5 " " }{XPPEDIT 262 1 "diff(y(t),t) = 16*y-2*y^2-x*y;" "6#/-%% diffG6$-%\"yG6#%\"tGF*,(*&\"#;\"\"\"F(F.F.*&\"\"#F.*$F(F0F.!\"\"*&%\"x GF.F(F.F2" }{TEXT 695 2 " " }}{PARA 3 "" 0 "" {TEXT 696 9 "that was \+ " }{TEXT 691 49 "given in problem 15 and considered in problem 16." }} {PARA 0 "" 0 "" {TEXT 538 0 "" }}{PARA 0 "" 0 "" {TEXT 539 15 "a) noda l source" }}{PARA 0 "" 0 "" {TEXT 540 13 "b) nodal sink" }}{PARA 0 "" 0 "" {TEXT 541 22 "c) stable saddle point" }}{PARA 0 "" 0 "" {TEXT 542 48 "d) stable and asymptotically stable saddle point" }}{PARA 0 " " 0 "" {TEXT 543 24 "e) unstable saddle point" }}{PARA 0 "" 0 "" {TEXT 544 14 "f) stable star" }}{PARA 0 "" 0 "" {TEXT 545 16 "g) unsta ble star" }}{PARA 0 "" 0 "" {TEXT 546 16 "h) stable center" }}{PARA 0 "" 0 "" {TEXT 547 24 "i) unstable spiral point" }}{PARA 0 "" 0 "" {TEXT 548 37 "j) asymptotically stable spiral point" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 801 9 "Solution:" }{TEXT -1 2 " " }{TEXT 802 5 "( e )" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}{PARA 7 "" 1 "" {TEXT 375 45 "Warning, the name adjoint has been redefined\n " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "eigenvals(matrix([[6, 0 ], [-8, -16]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"'!#;" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 803 115 "The rea l eigenvalues of opposite sign tell us that the cp is a saddle point. \+ There are only unstable saddle points." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 524 13 "18. Suppose " }{XPPEDIT 525 1 "f(t);" "6#-%\"fG6#%\"tG" }{TEXT 526 20 " that equals 1 for " } {XPPEDIT 527 1 "t;" "6#%\"tG" }{TEXT 528 57 " in the interval [1,2) a nd 2 in the interval (2,3]. Let " }{XPPEDIT 529 1 "F(t);" "6#-%\"FG6#% \"tG" }{TEXT 530 26 " be the Fourier series of " }{XPPEDIT 531 1 "f;" "6#%\"fG" }{TEXT 532 12 ". What is " }{XPPEDIT 533 1 "F(3);" "6#-%\" FG6#\"\"$" }{TEXT 534 2 " ?" }}{PARA 0 "" 0 "" {TEXT 535 0 "" }}{PARA 0 "" 0 "" {TEXT 536 104 "a) 0 b) 1/2 c) 1 d) 3/2 e) 2 f) 5/2 g) 3 h)7/2 i) 4 j) undefined " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 729 9 "Solution:" } {TEXT -1 1 "\000" }{TEXT 730 3 " " }{TEXT 731 7 "( d ) " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 732 5 "Let " }{XPPEDIT 19 1 "phi;" "6#%$ phiG" }{TEXT 736 31 " denote the periodization of " }{XPPEDIT 19 1 " f;" "6#%\"fG" }{TEXT 737 81 ", adjusted at jump discontinuities to be the average of the one-sided limits of " }{XPPEDIT 19 1 "f;" "6#%\"fG " }{TEXT 738 8 ". Thus, " }{XPPEDIT 19 1 "phi(3) = (limit(f(t),t = 3,l eft)+limit(f(t),t = 3,right))/2;" "6#/-%$phiG6#\"\"$*&,&-%&limitG6%-% \"fG6#%\"tG/F0F'%%leftG\"\"\"-F+6%-F.6#F0/F0F'%&rightGF3F3\"\"#!\"\"" }{TEXT 739 9 ". Thus, " }{XPPEDIT 19 1 "phi(3) = (limit(f(t),t = 3,le ft)+limit(f(t),t = 1,right))/2;" "6#/-%$phiG6#\"\"$*&,&-%&limitG6%-%\" fG6#%\"tG/F0F'%%leftG\"\"\"-F+6%-F.6#F0/F0F3%&rightGF3F3\"\"#!\"\"" } {TEXT 740 14 ". This value, " }{XPPEDIT 19 1 "3/2;" "6#*&\"\"$\"\"\"\" \"#!\"\"" }{TEXT 741 20 ", is the value that " }{XPPEDIT 19 1 "F(3);" "6#-%\"FG6#\"\"$" }{TEXT 742 13 "converges to." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 429 8 "19. Let " }{XPPEDIT 430 1 "f;" "6#%\"fG" }{TEXT 431 2 " " }{MPLTEXT 1 0 0 "" }{TEXT 432 149 "be the periodization (of per iod 2) of the function that is -1 on the interval [-1,0) and 1 on the \+ interval (0,1]. Which of the following statements " }{TEXT 428 4 "bes t" }{TEXT 433 52 " describes the Fourier series\n \+ " }{XPPEDIT 434 1 "a[0]/2+Sum(a[k]*cos(k*Pi*t)+b[k]*sin(k*Pi*t),k = 1 .. infinity);" "6#,&*&&%\"aG6#\"\"!\"\"\"\"\"#!\"\"F)-%$SumG6$,&*&&F& 6#%\"kGF)-%$cosG6#*(F3F)%#PiGF)%\"tGF)F)F)*&&%\"bG6#F3F)-%$sinG6#*(F3F )F8F)F9F)F)F)/F3;F)%)infinityGF)" }{TEXT 435 1 " " }}{PARA 3 "" 0 "" {TEXT 436 3 "of " }{XPPEDIT 437 1 "f;" "6#%\"fG" }{TEXT 438 215 " ? (R ead all proposed answers before choosing since more than one answer ma y be true. If that is the case, then choose the answer that conveys th e greatest amount of true information about the Fourier coefficients.) " }}{PARA 0 "" 0 "" {TEXT 439 0 "" }}{PARA 0 "" 0 "" {TEXT 440 3 "a) \+ " }{XPPEDIT 441 1 "a[k] = 0;" "6#/&%\"aG6#%\"kG\"\"!" }{TEXT 442 10 " \+ for all " }{XPPEDIT 443 1 "0 <= k;" "6#1\"\"!%\"kG" }}{PARA 0 "" 0 " " {TEXT 444 3 "b) " }{XPPEDIT 445 1 "a[k] = 0;" "6#/&%\"aG6#%\"kG\"\"! " }{TEXT 446 10 " for all " }{XPPEDIT 447 1 "0 < k;" "6#2\"\"!%\"kG" }}{PARA 0 "" 0 "" {TEXT 448 3 "c) " }{XPPEDIT 449 1 "b[k] = 0;" "6#/&% \"bG6#%\"kG\"\"!" }{TEXT 450 10 " for all " }{XPPEDIT 451 1 "0 <= k; " "6#1\"\"!%\"kG" }}{PARA 0 "" 0 "" {TEXT 452 3 "d) " }{XPPEDIT 453 1 "b[k] = 0;" "6#/&%\"bG6#%\"kG\"\"!" }{TEXT 454 24 " for all positive \+ even " }{XPPEDIT 455 1 "k;" "6#%\"kG" }}{PARA 0 "" 0 "" {TEXT 456 3 "e ) " }{XPPEDIT 457 1 "b[k] = 0;" "6#/&%\"bG6#%\"kG\"\"!" }{TEXT 458 23 " for all positive odd " }{XPPEDIT 459 1 "k;" "6#%\"kG" }}{PARA 0 "" 0 "" {TEXT 460 3 "f) " }{XPPEDIT 461 1 "a[k] = 0;" "6#/&%\"aG6#%\"kG\" \"!" }{TEXT 462 25 " for all positive even " }{XPPEDIT 463 1 "k;" "6 #%\"kG" }{TEXT 464 6 " and " }{XPPEDIT 465 1 "b[k] = 0;" "6#/&%\"bG6# %\"kG\"\"!" }{TEXT 466 22 " for all positive odd " }{XPPEDIT 467 1 "k; " "6#%\"kG" }}{PARA 0 "" 0 "" {TEXT 468 3 "g) " }{XPPEDIT 469 1 "b[k] \+ = 0;" "6#/&%\"bG6#%\"kG\"\"!" }{TEXT 470 10 " for all " }{XPPEDIT 471 1 "0 <= k;" "6#1\"\"!%\"kG" }{TEXT 472 6 " and " }{XPPEDIT 473 1 "a[k] = 0;" "6#/&%\"aG6#%\"kG\"\"!" }{TEXT 474 23 " for all positive e ven " }{XPPEDIT 475 1 "k;" "6#%\"kG" }}{PARA 0 "" 0 "" {TEXT 476 3 "h) " }{XPPEDIT 477 1 "b[k] = 0;" "6#/&%\"bG6#%\"kG\"\"!" }{TEXT 478 10 " for all " }{XPPEDIT 479 1 "0 <= k;" "6#1\"\"!%\"kG" }{TEXT 480 6 " \+ and " }{XPPEDIT 481 1 "a[k] = 0;" "6#/&%\"aG6#%\"kG\"\"!" }{TEXT 482 22 " for all positive odd " }{XPPEDIT 483 1 "k;" "6#%\"kG" }}{PARA 0 " " 0 "" {TEXT 484 3 "i) " }{XPPEDIT 485 1 "a[k] = 0;" "6#/&%\"aG6#%\"kG \"\"!" }{TEXT 486 10 " for all " }{XPPEDIT 487 1 "0 <= k;" "6#1\"\"!% \"kG" }{TEXT 488 6 " and " }{XPPEDIT 489 1 "b[k] = 0;" "6#/&%\"bG6#% \"kG\"\"!" }{TEXT 490 23 " for all positive even " }{XPPEDIT 491 1 "k; " "6#%\"kG" }}{PARA 0 "" 0 "" {TEXT 492 3 "j) " }{XPPEDIT 493 1 "a[k] \+ = 0;" "6#/&%\"aG6#%\"kG\"\"!" }{TEXT 494 10 " for all " }{XPPEDIT 495 1 "0 <= k;" "6#1\"\"!%\"kG" }{TEXT 496 6 " and " }{XPPEDIT 497 1 "b[k] = 0;" "6#/&%\"bG6#%\"kG\"\"!" }{TEXT 498 22 " for all positive o dd " }{XPPEDIT 499 1 "k;" "6#%\"kG" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 724 9 "Solution:" }{TEXT -1 1 "\000" }{TEXT 725 3 " " }{TEXT 726 7 "( i ) " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 728 7 "Because" }{TEXT -1 2 " " }{XPPEDIT 19 1 "f;" "6#%\"fG" }{TEXT -1 2 " " }{TEXT 727 68 "is an odd function its Fourier series is a Fo urier sine series, i.e." }{TEXT -1 1 " " }{XPPEDIT 19 1 "a[k] = 0;" "6 #/&%\"aG6#%\"kG\"\"!" }{TEXT -1 2 " " }{TEXT 735 7 "for all" }{TEXT -1 1 " " }{XPPEDIT 19 1 "k;" "6#%\"kG" }{TEXT -1 3 ".\n " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "b[k] = int(-1*sin(Pi*k*t), t=-1..0) + int(1*sin(Pi*k*t), t=0..1); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"bG6#%\"kG,$*&,&!\"\"\"\"\"-%$cosG6#*&%#PiGF,F'F,F,F,*&F1F,F'F,F+!\"# " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 733 9 "Thus , for" }{TEXT -1 1 " " }{XPPEDIT 19 1 "b[k] = 0;" "6#/&%\"bG6#%\"kG\" \"!" }{TEXT -1 1 " " }{TEXT 734 4 "even" }{TEXT -1 1 " " }{XPPEDIT 19 1 "k;" "6#%\"kG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 0 {PARA 0 "" 0 "" {TEXT 502 9 "20. Let " }{XPPEDIT 503 1 "f(t) \+ = Sum(sin(n*t)/(n^2),n = 1 .. infinity);" "6#/-%\"fG6#%\"tG-%$SumG6$*& -%$sinG6#*&%\"nG\"\"\"F'F1F1*$F0\"\"#!\"\"/F0;F1%)infinityG" }{TEXT 504 126 " . Which of the following expressions is the Fourier series \+ expansion of a particular solution of the differential equation " } {XPPEDIT 257 0 "diff(x(t),`$`(t,2))+Pi^2*x(t) = f(t);" "6#/,&-%%diffG6 $-%\"xG6#%\"tG-%\"$G6$F+\"\"#\"\"\"*&%#PiGF/-F)6#F+F0F0-%\"fG6#F+" } {TEXT 523 1 "?" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 505 3 "a) " } {XPPEDIT 697 1 "1/(2*Pi)+sum((cos(n*t)+sin(n*t))/(Pi^2*n^2),n = 1 .. i nfinity);" "6#,&*&\"\"\"F%*&\"\"#F%%#PiGF%!\"\"F%-%$sumG6$*&,&-%$cosG6 #*&%\"nGF%%\"tGF%F%-%$sinG6#*&F3F%F4F%F%F%*&F(F'F3F'F)/F3;F%%)infinity GF%" }}{PARA 0 "" 0 "" {TEXT 506 3 "b) " }{XPPEDIT 507 1 "1/(2*Pi)+sum (cos(n*t)/(Pi^2*n^2),n = 1 .. infinity);" "6#,&*&\"\"\"F%*&\"\"#F%%#Pi GF%!\"\"F%-%$sumG6$*&-%$cosG6#*&%\"nGF%%\"tGF%F%*&F(F'F2F'F)/F2;F%%)in finityGF%" }}{PARA 0 "" 0 "" {TEXT 508 3 "c) " }{XPPEDIT 509 1 "sum(si n(n*t)/(Pi^2*n^2),n = 1 .. infinity);" "6#-%$sumG6$*&-%$sinG6#*&%\"nG \"\"\"%\"tGF,F,*&%#PiG\"\"#F+F0!\"\"/F+;F,%)infinityG" }}{PARA 0 "" 0 "" {TEXT 510 5 "d) " }{XPPEDIT 511 1 "1/(2*Pi)+sum((cos(n*t)+sin(n*t ))/((Pi^2-n^2)*n^2),n = 1 .. infinity);" "6#,&*&\"\"\"F%*&\"\"#F%%#PiG F%!\"\"F%-%$sumG6$*&,&-%$cosG6#*&%\"nGF%%\"tGF%F%-%$sinG6#*&F3F%F4F%F% F%*&,&*$F(F'F%*$F3F'F)F%*$F3F'F%F)/F3;F%%)infinityGF%" }}{PARA 0 "" 0 "" {TEXT 512 3 "e) " }{XPPEDIT 513 1 "1/(2*Pi)+sum(cos(n*t)/((Pi^2-n^2 )*n^2),n = 1 .. infinity);" "6#,&*&\"\"\"F%*&\"\"#F%%#PiGF%!\"\"F%-%$s umG6$*&-%$cosG6#*&%\"nGF%%\"tGF%F%*&,&*$F(F'F%*$F2F'F)F%*$F2F'F%F)/F2; F%%)infinityGF%" }}{PARA 0 "" 0 "" {TEXT 514 3 "f) " }{XPPEDIT 515 1 " sum(sin(n*t)/((Pi^2-n^2)*n^2),n = 1 .. infinity);" "6#-%$sumG6$*&-%$si nG6#*&%\"nG\"\"\"%\"tGF,F,*&,&*$%#PiG\"\"#F,*$F+F2!\"\"F,*$F+F2F,F4/F+ ;F,%)infinityG" }}{PARA 0 "" 0 "" {TEXT 516 3 "g) " }{XPPEDIT 517 1 "1 /(2*Pi)+sum(cos(n*t)/(Pi^2+n^2),n = 1 .. infinity);" "6#,&*&\"\"\"F%*& \"\"#F%%#PiGF%!\"\"F%-%$sumG6$*&-%$cosG6#*&%\"nGF%%\"tGF%F%,&*$F(F'F%* $F2F'F%F)/F2;F%%)infinityGF%" }}{PARA 0 "" 0 "" {TEXT 518 3 "h) " } {XPPEDIT 519 1 "sum(sin(n*t)/(Pi^2-n^2),n = 1 .. infinity);" "6#-%$sum G6$*&-%$sinG6#*&%\"nG\"\"\"%\"tGF,F,,&*$%#PiG\"\"#F,*$F+F1!\"\"F3/F+;F ,%)infinityG" }}{PARA 0 "" 0 "" {TEXT 520 3 "i) " }{XPPEDIT 521 1 "Sum (sin(n*sqrt(Pi)*t)/(n^2),n = 1 .. infinity)" "6#-%$SumG6$*&-%$sinG6#*( %\"nG\"\"\"-%%sqrtG6#%#PiGF,%\"tGF,F,*$F+\"\"#!\"\"/F+;F,%)infinityG" }}{PARA 0 "" 0 "" {TEXT 522 3 "j) " }{XPPEDIT 708 1 "Sum(sin(n*Pi*t)/( n^2),n = 1 .. infinity)" "6#-%$SumG6$*&-%$sinG6#*(%\"nG\"\"\"%#PiGF,% \"tGF,F,*$F+\"\"#!\"\"/F+;F,%)infinityG" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 719 9 "Solution:" }{TEXT -1 1 "\000" } {TEXT 720 3 " " }{TEXT 723 6 "( f )" }}{PARA 0 "" 0 "" {TEXT 722 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "subs(x(t) = sum(b[n]*sin(n*t),n=1..infinity), diff(x (t), t$2)+Pi^2*x(t) = sum(sin(n*t)/(n^2),n = 1 .. infinity));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-%%diffG6$-%$sumG6$*&&%\"bG6#%\"nG \"\"\"-%$sinG6#*&F/F0%\"tGF0F0/F/;F0%)infinityG-%\"$G6$F5\"\"#F0*&)%#P iGF " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-% $sumG6$,$*(&%\"bG6#%\"nG\"\"\"-%$sinG6#*&F-F.%\"tGF.F.)F-\"\"#F.!\"\"/ F-;F.%)infinityGF.*&)%#PiGF5F.-F&6$*&F*F.F/F.F7F.F.-F&6$*&F/F.*$F4F.F6 F7" }}}{PARA 0 "" 0 "" {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 721 85 "By equating coefficients of similar sine terms on each side of the eq uation we obtain" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 30 "-b[n]*n^2 + Pi^2*b[n] = 1/n^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&&%\"bG6#%\"nG\"\"\")F)\"\"#F*!\"\"*&)%#PiGF,F *F&F*F**&F*F**$F+F*F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "so lve(%, b[n]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"\"F%*&)%\"nG \"\"#F%,&*$F'F%F%*$)%#PiGF)F%!\"\"F%F/F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 376 18 "Laplace Transforms" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 " \+ In this table," }{TEXT 686 2 " " }{XPPEDIT 256 1 "F(s )" "6#-%\"FG6#%\"sG" }{TEXT 687 1 " " }{TEXT -1 35 " denotes the Lapl ace transform of " }{TEXT 688 1 " " }{XPPEDIT 257 1 "f(t)" "6#-%\"fG6# %\"tG" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 25 " \+ " }{TEXT 684 10 "Expression" }{TEXT -1 36 " \+ " }{TEXT 685 17 "Laplace Transform" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT 377 24 " \+ " }{XPPEDIT 855 1 "t^p" "6#)%\"tG%\"pG" }{TEXT 856 47 " " }{XPPEDIT 857 1 " GAMMA(p+1)/s^(p+1)" "6#*&-%&GAMMAG6#,&%\"pG\"\"\"F)F)F))%\"sG,&F(F)F)F )!\"\"" }}{PARA 258 "" 0 "" {TEXT 858 11 " " }}{PARA 258 "" 0 "" {TEXT 859 23 " " }{XPPEDIT 860 1 "exp(k*t) " "6#-%$expG6#*&%\"kG\"\"\"%\"tGF(" }{TEXT 861 45 " \+ " }{XPPEDIT 862 1 "1/(s-k)" "6#*&\"\"\"F$,&% \"sGF$%\"kG!\"\"F(" }}{PARA 258 "" 0 "" {TEXT 863 7 " " }}{PARA 258 "" 0 "" {TEXT 864 21 " " }{XPPEDIT 865 1 "cos( omega*t)" "6#-%$cosG6#*&%&omegaG\"\"\"%\"tGF(" }{TEXT 866 40 " \+ " }{XPPEDIT 867 1 "s/(s^2+omega^2)" "6 #*&%\"sG\"\"\",&*$F$\"\"#F%*$%&omegaGF(F%!\"\"" }{TEXT 868 4 " " }} {PARA 258 "" 0 "" {TEXT 869 0 "" }}{PARA 258 "" 0 "" {TEXT 870 22 " \+ " }{XPPEDIT 871 1 "sin(omega*t)" "6#-%$sinG6#*&%&om egaG\"\"\"%\"tGF(" }{TEXT 872 39 " \+ " }{XPPEDIT 873 1 "omega/(s^2+omega^2)" "6#*&%&omegaG\"\"\",&*$%\"s G\"\"#F%*$F$F)F%!\"\"" }}{PARA 258 "" 0 "" {TEXT 874 0 "" }}{PARA 258 "" 0 "" {TEXT 875 22 " " }{XPPEDIT 876 1 "D(f)(t) " "6#--%\"DG6#%\"fG6#%\"tG" }{TEXT 877 36 " \+ " }{XPPEDIT 878 1 "s*F(s)-f(0)" "6#,&*&%\"sG\"\"\"-%\"FG6#F%F &F&-%\"fG6#\"\"!!\"\"" }{TEXT 879 5 " " }}{PARA 258 "" 0 "" {TEXT 880 0 "" }}{PARA 258 "" 0 "" {TEXT 881 20 " " } {XPPEDIT 882 1 "(D@@2)(f)(t)" "6#---%#@@G6$%\"DG\"\"#6#%\"fG6#%\"tG" } {TEXT 883 23 " " }{XPPEDIT 884 1 "s^2*F(s)-s*f(0 )-D(f)(0)" "6#,(*&%\"sG\"\"#-%\"FG6#F%\"\"\"F**&F%F*-%\"fG6#\"\"!F*!\" \"--%\"DG6#F-6#F/F0" }{TEXT 885 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 258 "" 0 "" {TEXT 886 0 "" }}{PARA 258 "" 0 "" {TEXT 887 22 " " }{XPPEDIT 888 1 "exp(a*t)*f(t)" "6#*&-%$expG6 #*&%\"aG\"\"\"%\"tGF)F)-%\"fG6#F*F)" }{TEXT 889 35 " \+ " }{XPPEDIT 890 1 "F(s-a)" "6#-%\"FG6#,&%\"sG\"\"\"% \"aG!\"\"" }}{PARA 258 "" 0 "" {TEXT 891 3 " " }{TEXT -1 0 "" }}}} {MARK "21 21 0" 3 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }