{VERSION 2 3 "APPLE_PPC_MAC" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 256 "" 0 "" {TEXT -1 18 "Chaper 2 Section 4" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 105 "In this section, \+ we have two-point mixed problems. The text constructs a Green's functi on for the problem" }}{PARA 0 "" 0 "" {TEXT -1 66 " L(y) = y \+ ' ', y(0) + y(1) = 0, y ' (0) +y ' (1) = 0. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 9 "Exercises" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "I. We find a soluti on for the problem" }}{PARA 0 "" 0 "" {TEXT -1 19 " y ' ' = \+ " }{XPPEDIT 18 0 "x^2" "*$%\"xG\"\"#" }{TEXT -1 46 ", y(0) + y(1) \+ = 0, y ' (0) +y ' (1) = 0. " }}{PARA 0 "" 0 "" {TEXT -1 50 "The Green s function was computed for this example." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "G:=(x,t)->piecewise(x " 0 "" {MPLTEXT 1 0 31 "assume(x>0): additiona lly(x<1):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "int(G(x,t)*t^2 ,t=0..1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "y:=unapply(\", x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "y(0)+y(1); D(y)(0)+D (y)(1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "plot(y(x),x=0..1 );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 87 "2. In this second problem, we use Maple \+ to solve the four equations with four unknowns." }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 22 "restart: with(linalg):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 42 "deq:=diff(y(x),x,x)+3*diff(y(x),x)+2*y(x);" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "dsolve(deq,y(x),output=basi s);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "G:=(x,t)->piecewise( x " 0 "" {MPLTEXT 1 0 30 "assume(0 " 0 "" {MPLTEXT 1 0 21 "eq1:=G(0,t)-G(1,t)=0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "eq2:=D[1](G)(0,t)-D[1](G)(1,t)=0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "eq3:=c*exp(-2*t)+d*exp(-t )=a*exp(-2*t)+b*exp(-t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "eq4:=(-2*c*exp(-2*t)-d*exp(-t))-(-2*a*exp(-2*t)-b*exp(-t))=1;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "solve(\{eq1,eq2,eq3,eq4\},\{ a,b,c,d\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "assign(\"); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "plot3d(G(x,t),x=0..1,t= 0..1,axes=NORMAL);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "assum e(x>0): additionally(x<1):\nint(G(x,t)*1,t=0..1);" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 22 "expand(\");simplify(\");" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "y:=unapply(\",x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "plot(y(x),x=0..1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "diff(y(x),x,x)+3*diff(y(x),x)+2*y(x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "assume(x>0): additionally(x<1):\nin t(G(x,t)*sin(Pi*t),t=0..1):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "expand(\"):simplify(\");" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "y:=unapply(\",x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 " plot(y(x),x=0..1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {MARK "0 0" 16 }{VIEWOPTS 1 1 0 1 1 1803 }