{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 }{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 }} {SECT 0 {PARA 0 "" 0 "" {TEXT -1 21 "Chapter 2 Section 4." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 467 " In this par t of Section 4, we create G for the problem L(y) = f, where y satisfie d boundary conditions. The text explains that G(x,t) satisfies L(G(x,t )) = 0 as a function of x. For second order problems, the method reduc es to four equations. The coefficients for G are determined by these f our equations. the first two equations are the boundary conditions. Th e third equation is the continuity equations, and the last equation is the \" jump\" in the derivative." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 9 "Exercises" }}{PARA 0 "" 0 "" {TEXT -1 56 " Exercise I. In each of these, M = \{y: y(0) = y'(0) = 0\}." }}{PARA 0 "" 0 "" {TEXT -1 14 "Ia L(y) = y''" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "dsolve(diff(y(x),x,x)=0,y(x),output=basis);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "u:=x->a*x+b;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "eq1:=b=0; eq2:=a=0; eq3:=a*t+b=c*t+d; eq4:=c-a=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 37 "G:=(x,t)->piecewise(x< t,a*x+b,c*x+d);" }}}{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 "assume(x>0); additionally(x<1);\nint(G(x,t)*t,t=0..1) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "c*x+d;" }}}{PARA 0 "" 0 "" {TEXT -1 27 "1b. L(y) = y'' + 4*¹¥¹ y(x)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "dsolve(diff(y(x),x,x)+4*Pi^2*y=0,y(x),output=basis);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "u:=x->a*cos(2*Pi*x)+b*sin(2*Pi*x); \nv:=x->c*cos(2*Pi*x)+d*sin(2*Pi*x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "eq1:=u(0)=0; eq2:=D(u)(0)=0; eq3:=v(t)-u(t)=0; eq4:=D (v)(t)-D(u)(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 35 "G:=(x,t)->piecewise(x " 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 "assume(x>0); additionally(x< 1);\nint(G(x,t)*t,t=0..1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "y:=unapply(\",x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "pl ot(y(x),x=0..1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "diff(y( x),x,x)+4*Pi^2*y(x):\nsimplify(\",trig);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "combine(v(x),trig);" }}}{PARA 0 "" 0 "" {TEXT -1 25 " 1c. L(y) = 2*y'' + y' - y" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 " restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "dsolve(2*diff(y (x),x,x)+diff(y(x),x)-y=0,y(x),output=basis);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "u:=x->a*exp(-x)+b*exp(x/2);\nv:=x->c*exp(-x)+d*e xp(x/2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "eq1:=u(0)=0; eq 2:=D(u)(0)=0; eq3:=v(t)-u(t)=0; eq4:=D(v)(t)-D(u)(t)=1/2;" }}}{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 35 "G:=(x,t)->piecewise(x " 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 "assume(x>0); additionally(x<1);\nint(G(x,t)*t,t=0..1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "expand(\");" }}}{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 35 "2*diff(y(x),x,x)+diff(y(x),x)-y(x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "combine(v(x),exp);" }}}{PARA 0 "" 0 "" {TEXT -1 33 "1d. L(y) = exp(x) y'' + exp(x) y'" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "dsolve(exp(x)*diff(y(x),x,x)+exp(x)*diff(y(x),x)=0,y(x),output=b asis);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "u:=x->a+b*exp(-x) ;\nv:=x->c+d*exp(-x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "eq 1:=u(0)=0; eq2:=D(u)(0)=0; eq3:=v(t)-u(t)=0; eq4:=D(v)(t)-D(u)(t)=1/ex p(t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "solve(\{eq1,eq2,eq 3,eq4\},\{a,b,c,d\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "as sign(\");" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "G:=(x,t)->piec ewise(x " 0 "" {MPLTEXT 1 0 41 "pl ot3d(G(x,t),x=0..1,t=0..1,axes=NORMAL);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "assume(x>0); additionally(x<1);\nint(G(x,t)*t,t=0..1) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "expand(\");" }}} {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 28 "diff(exp(x)*diff(y(x),x),x); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "v(x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0" 21 }{VIEWOPTS 1 1 0 1 1 1803 }