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{MPLTEXT 1 0 40 "W:=wronskian([v0(x),v1(x)],x);W:=det(W);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%\"WG-%'matrixG6#7$7$*&%\"AG\"\"\"-%$sinG6#%\"
xGF,,&*(%\"BGF,-F.6#F,F,-%$cosGF/F,!\"\"*(F3\"\"\"-F7F5F,F-F:F,7$*&F+F
:F6F:,&*(F3F:F4F:F-F:F,*(F3F:F;F:F6F:F," }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"WG,&**%\"AG\"\"\")-%$sinG6#%\"xG\"\"#\"\"\"%\"BGF(-F+6#F(F(F
(**F'F/)-%$cosGF,F.F/F0F/F1F/F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 12 "simplify(W);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(%\"AG\"\"\"%
\"BGF%-%$sinG6#F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "A:=1
;B:=-1/sin(1);simplify(W);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG\"
\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG,$*&\"\"\"F'-%$sinG6#\"
\"\"!\"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "Grf(x,y);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%*PIECEWISEG6$7$*&,&-%$cosG6#%\"xG\"\"\"*&*&-F*6#F-F--
%$sinGF+F-\"\"\"-F3F1!\"\"!\"\"F--F36#%\"yGF-31,$F:F7\"\"!1,&F:F-F,F7F
>7$*&F2F4,&-F*F9F-*&*&F0F4F8F4F4F5F6F7F-31,&F,F-F:F7F>1,&F:F-F7F-F>" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "This is the result. Let's compar
e it with the one given on probl. 2.8." }}{PARA 0 "" 0 "" {TEXT -1 1 "
 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "G1:=c*sin(y)*sin(1-x);
G2:=c*sin(x)*sin(1-y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#G1G,$*(%
\"cG\"\"\"-%$sinG6#%\"yGF(-F*6#,&!\"\"F(%\"xGF(F(F0" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%#G2G,$*(%\"cG\"\"\"-%$sinG6#%\"xGF(-F*6#,&%\"yGF(!
\"\"F(F(F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "expand(G1,tri
g):subs(c=1/sin(1),%);expand(G2,trig):subs(c=1/sin(1),%);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#,&*&-%$sinG6#%\"yG\"\"\"-%$cosG6#%\"xGF)F)*&*(F
%\"\"\"-F+6#F)F)-F&F,F)F0-F&F2!\"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#,&*&*(-%$sinG6#%\"yG\"\"\"-%$cosG6#F*F*-F'6#%\"xGF*\"\"\"-F'F-!
\"\"!\"\"*&F.F1-F,F(F*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "Compa
ring these formulas we see that they do coincide." }}{PARA 0 "" 0 "" 
{TEXT -1 26 "Let's finally make a plot:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 31 "plot3d(Grf(x,y),x=0..1,y=0..1);" }}{PARA 13 "" 1 "" {INLPLOT "6$
-%%GRIDG6%;\"\"!$\"\"\"F'F&W(\\bm\":\":0000000000000000000000000000000
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0 0 0 0 0 0 0 0 1 0 0 0 0 2121 0 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 72 "f:=x->x^2;\n assume(x>0);additionally(x<1);\n u:=i
nt(G(x,y)*f(y),y=0..1);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#
%\"xG6\"6$%)operatorG%&arrowGF(*$)9$\"\"#\"\"\"F(F(F(" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%\"uG,&*$)%#x|irG\"\"%\"\"\"#!\"\"\"#7F(#\"\"\"F-
" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 8 "Exe 2.14" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "restart:with
(linalg):read(\"/home/www/public_html/teaching/osdylask/maple/ohjelmat
.mpl\"):" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning, new definition for n
orm" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definition for trace
" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 10 "Scheme S1:" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 76 "n:=7:A:=matrix(n,n,(i,j)->piecewise(abs(i
-j)=1,-1,i=j,2,0)):A[n,n]:=1:op(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#-%'matrixG6#7)7)\"\"#!\"\"\"\"!F*F*F*F*7)F)F(F)F*F*F*F*7)F*F)F(F)F*F*
F*7)F*F*F)F(F)F*F*7)F*F*F*F)F(F)F*7)F*F*F*F*F)F(F)7)F*F*F*F*F*F)\"\"\"
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "X0:=[seq(k*h,k=0..n+1)]
;X:=X0[1..n+1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#X0G7+\"\"!%\"hG,
$F'\"\"#,$F'\"\"$,$F'\"\"%,$F'\"\"&,$F'\"\"',$F'\"\"(,$F'\"\")" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"XG7*\"\"!%\"hG,$F'\"\"#,$F'\"\"$,$
F'\"\"%,$F'\"\"&,$F'\"\"',$F'\"\"(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
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rrowGF)-%*piecewiseG6&31\"\"!9%1F39$*&F3\"\"\",&F7F7F5!\"\"F731F5F31F3
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6$%\"xG%\"sG6\"6$%)operatorG%&arrowGF)-%*piecewiseG6&31\"\"!9%1F39$*&-
%#v1G6#F5\"\"\"-%#v0G6#F3F:31F5F31F3F:*&-F<F9F:-F8F=F:F)F)F)" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 21 "f:=x->100*exp(-10*x);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"fGR6#%\"xG6\"6$%)operatorG%&arrowGF(,$-%$expG6#,$9$!#5\"$+\"
F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "\n assume(x>0);ad
ditionally(x<1);\n " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "u:=s
implify(eval(int(G(x,y)*f(y),y=0..1)));\ndiff(u,x,x);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%\"uG,*\"\"\"F&*&%#x|irGF&-%$expG6#!#5F&F&F(!\"\"
-F*6#,$F(F,F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%$expG6#,$%#x|irG!
#5!$+\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "u:=unapply(u,x);
u(0),u(1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uGR6#%#x|irG6\"6$%)o
peratorG%&arrowGF(,*\"\"\"F-*&9$F--%$expG6#!#5F-F-F/!\"\"-F16#,$F/F3F4
F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"!F#" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 7 "x:='x':" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 5 "u(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*\"\"\"F$*&%
\"xGF$-%$expG6#!#5F$F$F&!\"\"-F(6#,$F&F*F+" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 30 "Diff(f(x),x,x)=diff(f(x),x,x);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%%DiffG6$,$-%$expG6#,$%\"xG!#5\"$+\"-%\"$G6$F,\"\"#,$
F(\"&++\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 38 "Let's do the numerical part in Matlab." }
}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 0 "" }}}}{MARK "5" 0 }{VIEWOPTS 1 
1 0 1 1 1803 }