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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "# file ~liska/vyuka
/odes/rk2\n" }{MPLTEXT 1 0 2 "#\n" }{MPLTEXT 1 0 33 "# Derivation of R
K2 coefficients\n" }{MPLTEXT 1 0 2 "#\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "f0 := f(x0,y(x0));\n" }{MPLTEXT 1 0 1 "\n" }}{PARA 11
 "" 1 "" {XPPMATH 20 ">I#f0G6\"-I\"fGF$6$I#x0GF$-I\"yGF$6#F(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "k1 := f0;\n" }{MPLTEXT 1 0 
1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#k1G6\"-I\"fGF$6$I#x0GF$-I\"y
GF$6#F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "k2 := f(x0+c2*h,
y(x0)+h*a21*k1);\n" }{MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 
20 ">I#k2G6\"-I\"fGF$6$,&*&I#c2GF$\"\"\"I\"hGF$F+F+I#x0GF$F+,&-I\"yGF$
6#F-F+*(F,F+I$a21GF$F+-F&6$F-F/F+F+" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 43 "fi2 := y(x0+h) - y(x0) -h*(b1*k1 + b2*k2);\n" }
{MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I$fi2G6\",(-I\"yG
F$6#,&I#x0GF$\"\"\"I\"hGF$F+F+-F'6#F*!\"\"*&F,F+,&*&I#b1GF$F+-I\"fGF$6
$F*F-F+F+*&I#b2GF$F+-F56$,&*&I#c2GF$F+F,F+F+F*F+,&F-F+*(F,F+I$a21GF$F+
F4F+F+F+F+F+F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "dfi2 := d
iff(fi2,h);\n" }{MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I
%dfi2G6\",*--I\"DG6$%*protectedGI(_syslibGF$6#I\"yGF$6#,&I#x0GF$\"\"\"
I\"hGF$F1F1*&I#b1GF$F1-I\"fGF$6$F0-F-6#F0F1!\"\"*&I#b2GF$F1-F66$,&*&I#
c2GF$F1F2F1F1F0F1,&F8F1*(F2F1I$a21GF$F1F5F1F1F1F:*(F2F1F<F1,&*&--&F(6#
F16#F6F>F1FAF1F1*(--&F(6#\"\"#FLF>F1FDF1F5F1F1F1F:" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 39 "dfi20 := subs(h=0,D(y)(x0) = f0,dfi2);\n" }
{MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I&dfi20G6\",(-I\"
fGF$6$I#x0GF$-I\"yGF$6#F)\"\"\"*&I#b1GF$F-F&F-!\"\"*&I#b2GF$F-F&F-F0" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "dfi20 := simplify(dfi20/f
0);\n" }{MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I&dfi20G6
\",(I#b1GF$!\"\"I#b2GF$F'\"\"\"F)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 23 "d2fi2 := diff(dfi2,h);\n" }{MPLTEXT 1 0 1 "\n" }}
{PARA 11 "" 1 "" {XPPMATH 20 ">I&d2fi2G6\",(---I#@@G6$%*protectedGI(_s
yslibGF$6$I\"DGF*\"\"#6#I\"yGF$6#,&I#x0GF$\"\"\"I\"hGF$F5F5*&I#b2GF$F5
,&*&--&F.6#F56#I\"fGF$6$,&*&I#c2GF$F5F6F5F5F4F5,&-F16#F4F5*(F6F5I$a21G
F$F5-F@6$F4FFF5F5F5FDF5F5*(--&F.6#F/F?FAF5FIF5FJF5F5F5!\"#*(F6F5F8F5,&
*&,&*&--&F.6$F5F5F?FAF5FDF5F5*(--&F.6$F5F/F?FAF5FIF5FJF5F5F5FDF5F5*(,&
*&FfnF5FDF5F5*(--&F.6$F/F/F?FAF5FIF5FJF5F5F5FIF5FJF5F5F5!\"\"" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "d2fi20 := simplify(\n" }
{MPLTEXT 1 0 25 "               subs(h=0,\n" }{MPLTEXT 1 0 30 "       \+
        D(y)(x0) = f0,\n" }{MPLTEXT 1 0 44 "               (D@@2)(y)(x
0) = diff(f0,x0),\n" }{MPLTEXT 1 0 24 "               d2fi2));\n" }
{MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I'd2fi20G6\",&*&,
&*(I#b2GF$\"\"\"I$a21GF$F*-I\"fGF$6$I#x0GF$-I\"yGF$6#F/F*!\"#--I\"DG6$
%*protectedGI(_syslibGF$6#F1F2F*F*--&F66#\"\"#6#F-F.F*F**&,&*&F)F*I#c2
GF$F*F3F*F*F*--&F66#F*F@F.F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 38 "d2fi20 := subs(D(y)(x0) = f0,d2fi20);\n" }{MPLTEXT 1 0 1 "\n" 
}}{PARA 11 "" 1 "" {XPPMATH 20 ">I'd2fi20G6\",&*&,&*(I#b2GF$\"\"\"I$a2
1GF$F*-I\"fGF$6$I#x0GF$-I\"yGF$6#F/F*!\"#F,F*F*--&I\"DG6$%*protectedGI
(_syslibGF$6#\"\"#6#F-F.F*F**&,&*&F)F*I#c2GF$F*F3F*F*F*--&F76#F*F=F.F*
F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "d2fi20 := collect(d2f
i20,\{f,D\});" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I'd2fi20G6\",&*&,&*&I#
b2GF$\"\"\"I#c2GF$F*!\"#F*F*F*--&I\"DG6$%*protectedGI(_syslibGF$6#F*6#
I\"fGF$6$I#x0GF$-I\"yGF$6#F8F*F**(,&*&I$a21GF$F*F)F*F,F*F*F*-F6F7F*--&
F06#\"\"#F5F7F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "co1 :=
 coeff(d2fi20,D[1](f)(x0, y(x0)),1);\n" }{MPLTEXT 1 0 1 "\n" }}{PARA 
11 "" 1 "" {XPPMATH 20 ">I$co1G6\",&*&I#b2GF$\"\"\"I#c2GF$F(!\"#F(F(" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "co2 := coeff(d2fi20,D[2](
f)(x0, y(x0)),1);\n" }{MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 
20 ">I$co2G6\"*&,&*&I$a21GF$\"\"\"I#b2GF$F)!\"#F)F)F)-I\"fGF$6$I#x0GF$
-I\"yGF$6#F/F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "co2 := si
mplify(co2/f0);\n" }{MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20
 ">I$co2G6\",&*&I$a21GF$\"\"\"I#b2GF$F(!\"#F(F(" }}}{EXCHG {PARA 0 "> 
" 0 "" {MPLTEXT 1 0 26 "eqs := \{dfi20, co1, co2\};\n" }{MPLTEXT 1 0 
1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I$eqsG6\"<%,(I#b1GF$!\"\"I#b2G
F$F(\"\"\"F*,&*&I$a21GF$F*F)F*!\"#F*F*,&*&F)F*I#c2GF$F*F.F*F*" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "sol := solve(eqs);\n" }
{MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I$solG6\"<&/I$a21
GF$,$*$I#b2GF$!\"\"#\"\"\"\"\"#/I#b1GF$,&F*F+F-F-/F*F*/I#c2GF$F(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "st := subs(sol,-(fi2-y(x0+h)
));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#stG6\",&-I\"yGF$6#I#x0GF$\"
\"\"*&I\"hGF$F*,&*&,&I#b2GF$!\"\"F*F*F*-I\"fGF$6$F)F&F*F**&F0F*-F36$,&
*&F0F1F,F*#F*\"\"#F)F*,&F&F**(F,F*F0F1F2F*F:F*F*F*F*" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 15 "subs(b2=1,sol);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "<&/\"\"\"F$/I$a21G6\"#F$\"\"#/I#b1GF'\"\"!/I#c2GF'F(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "subs(b2=1/2,sol);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "<&/#\"\"\"\"\"#F$/I$a21G6\"F%/I#b1GF)F$/I#c2GF
)F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 
"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> 
" 0 "" {MPLTEXT 1 0 0 "" }}}}
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