{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 "" -1 256 "" 0 1 0 0 0 0 0 1 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 }} {SECT 0 {PARA 0 "" 0 "" {TEXT 256 74 "Chapter 2 Section 5 Solving L(y) = Dirac( x - t ), with boundary condition" }{TEXT -1 1 "s" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 341 " The text in this section explains how to construct the Green's function using the Dirac Delta -- the unit pulse function. We use Maple to do this calcu lus. In each problem, we compute the Green's function as specified in \+ Exercise 1 of the section. It is used to construct a solution for the \+ ordinary differential equation of Exercise 2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 12 "Exercise 2.5" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "I and II (a) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "assume(t>0): additionally(t<1);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "dsolve(\{diff(y(x),x,x)+3*di ff(y(x),x)+2*y(x)=Dirac(x-t),\n y(0)=0,D(y)(0)=0\},y(x));" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "expand(rhs(\")): simplify(\" );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "G:=unapply(\",(x,t)); " }}}{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 55 "assum e(x>0): additionally(x<1):\nint(G(x,t)*t^2,t=0..1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "plot(\",x=0..1);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "I and II (b)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restar t:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "assume(t>0): addition ally(t<1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "dsolve(\{diff (y(x),x,x)+3*diff(y(x),x)+2*y(x)=Dirac(x-t),\n y(0)=0,y(1)=0\},y( x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "expand(rhs(\")): si mplify(\");" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "G:=unapply( \",(x,t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "plot3d(G(x,t) ,x=0..1,t=0..1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "assume( x>0): additionally(x<1):\nint(G(x,t)*1,t=0..1);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 23 "expand(\"): simplify(\");" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 15 "plot(\",x=0..1);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "I and II c" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "assume(t>0): additionally(t<1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "dsolve(diff(y(x),x,x)+3*diff(y(x),x)+2*y(x)=D irac(x-t),y(x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "G:=unap ply(rhs(\"),(x,t));" }}}{EXCHG {PARA 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 27 "solve(\{eq1,eq2\},\{_C1,_C2\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "assign(\"):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "assume(x>0): additionally(x<1):\nint(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 47 "simplify(diff(y(x),x,x)+3*di ff(y(x),x)+2*y(x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} }{MARK "2 0" 124 }{VIEWOPTS 1 1 0 1 1 1803 }