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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "restart: \n" }}}
{EXCHG {PARA 18 "" 0 "" {TEXT 256 22 "Transform\351e de Laplace" }
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "with(pl
dx): with(student): with(inttrans): with(plots, display):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 
-1 20 "Calculs \351l\351mentaires" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
20 "Fonction d'affichage" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
{TEXT -1 0 "" }{MPLTEXT 1 0 36 "vlap:= Lap=xlap: macro(H=Heaviside);" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "res1:= vlap(f)(p); \ndef_
lf:= (rhs=lhs)(vlap(f)(p));def_lg:= (rhs=lhs)(vlap(g)(p));" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "xcombipo:= proc(z); combine(z,power
,symbolic); end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "(xcombi
po@expand)(vlap(3*f+4*g)(p)); res2:= subs(def_lf, def_lg, 3=A, 4=B, %)
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "Lap:='Lap': define(Lap
, linear); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 505 "Lap:= proc(
)\nlocal theArgs, arg, look, me, cf, term;\n me := eval(procname, 1);
\n theArgs := args;\n arg := args[1];\n if arg = 1 then\n elif type(ar
g, 'name') then\n elif type(arg, 'constant') then return arg*procname(
1, args[2 .. -1])\n elif type(arg, '`+`') then return map(procname, ar
gs)\n elif type(arg, '`*`') then\n  cf, term := selectremove(type, arg
, 'constant');\n  if cf = 1 then return 'procname'(args) else return c
f*procname(term, args[2 .. -1]) end if\n end if;\n 'procname'(eval(the
Args,1))\nend proc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "Lap(
2*f+3*g); Lap(sin); Lap(t-> t^2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
9 "polyn\364mes" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "X:= t->t: vlap(1
)(p): xplif(%,[p,positive]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 34 "vlap(X)(p): xplif(%,[p,positive]);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 42 "vlap(eval(X^3))(p): xplif(%,[p,positive]);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "vlap(X^ 3/3!)(p): xplif(%,[p
,positive]);\nres3:= subs(3=k, -4=-(k+1), 1/6=1/k!, %);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 136 "p*xlap(f)(p); changevar(t*p=u,%,u)
; p*Lap(f)(p)=(xjoli@xplif)(%,[p, positive]);\nres31:= Limit(lhs(%),p=
infinity)=Limit(f(u),u=0, right);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 189 "p^2*xlap(f)(p); changevar(t*p=u,%,u); p^2*Lap(f)(p)=
(combine@xplif)(%,[p, positive]);\nf(u/p): %= mtaylor(%,u,2)+O(u^2)/p^
2; \nres32:= (xcombipo@xjoli@expand@value@xjoli@expand@subs)(%, %%%);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 189 "p^3*xlap(f)(p); change
var(t*p=u,%,u); p^3*Lap(f)(p)=(combine@xplif)(%,[p, positive]);\nf(u/p
): %= mtaylor(%,u,3)+O(u^3)/p^3; \nres33:= (xcombipo@xjoli@expand@valu
e@xjoli@expand@subs)(%, %%%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "exponentielle" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 67 "vlap(t->exp(a*t))(p): res4:= xplif(%,[p-a
,RealRange(0, infinity)]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
57 "vlap(exp@(I*id))(p): xplif(%,[p,RealRange(0, infinity)]);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "vlap(cos)(p): res5a:= xplif
(%,[p,positive], simplify);\nvlap(sin)(p): res5b:= xplif(%,[p,positive
], simplify);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "factor(r
es5a+I*res5b);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "retards" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 156 "fu:= (t->f(t-a)*H(t-a)); (xcombipo
@expand@changevar)(t-a=x, xlap(fu)(p),x);\nxplif(%, [a, positive]); re
s6:= Lap(eval(fu))(p)=subs(int=Int, x=t, def_lf, %); " }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "gra:= qqq-> displayf(plot(qqq,x=-5.
.10,linestyle=16, color=black), tickmarks=[3,3],axes=boxed):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "ff:= unapply(sin(x)*H(x),x);
 gra('ff'(x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "gg:= unap
ply(ff(t-2),t);  gra('gg'(x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 66 "hh:= unapply( (combine@simplify)(ff(t+2)*H(t)),t);   gra('hh'(
x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "xlap(ff)(p); (xcomb
ipo@normal@xplif)(%, [p, positive]);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 37 "xlap(gg)(p); xplif(%, [p, positive]);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "xlap(hh)(p); xplif(%, [p, positive]
);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "dilatation" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 147 "vlap(t->f(t/a))(p): subs(t=x, %); lhs(%)=(chang
evar)(x=t*a,rhs(%),t):\n(yjoli@xjoli@xplif)(%,[a,positive]);\nsubs(p=a
*p, def_lf): res7:= subs(%, %%);" }}}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 "D\351rivatio
n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "vlap(D(f))(p): \ndef_l
d:= convert(rhs(%), diff)=lhs(%);\n(yjoli@xjoli@xintparts)(%%, exp(-p*
t));\nres81:= (eval@subs)(limit=0, int=Int, def_lf, %); " }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "subs(f=D(f), res81, res81): res82:=
 collect(%, Lap);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "vlap(f
)(p); xjoli(diff(%,p)); \n(eval@subs)(f=(t->f(t)*t), def_lf): res9:= (
yjoli@subs)(%,%%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "vlap(t->f(t)/t)(p); (xjoli@
diff)(%,p);\n (yjoli@subs)((def_lf), %); map(Int,%,p=s..A); \nsubs(A=i
nfinity, value(-%)); res10:= combine(subsop([1,1]=0,%));" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 11 "Convolution" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 101 "Int(f(x)*g(t-x)*Heaviside(x)*Heaviside(t-x),x=0..t);
\nh:= unapply(subs(0..t=-infinity..infinity,%),t);" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 189 "subs(0=-infinity, xlap(h)(p)); tmp:= xfubi
ni(%); op(1,%): %=changevar(t=x+y,%,y):\n(xcombipo@expand@subs)(%, tmp
); \nres11:= Lap(f &* g)(p)=subs(int=Int, x=t, y=t, def_lf, def_lg, va
lue(%));\n" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 
{PARA 3 "" 0 "" {TEXT -1 16 "Elements simples" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }{TEXT -1 0 "" }{MPLTEXT 1 0 46 "(z^2+1)/z/(z-1
)/(z+2); convert(%, parfrac, z);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 40 "(2+z)/z^2/(z+1); convert(%, parfrac, z);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 8 "Equadiff" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "gv:= unapply(Heaviside(t)-He
aviside(t-1),t); def_gg:= g= gv;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 57 "lap_gg:= Lap(g)(p)= (xcombipo@expand@value)(xlap(gv)(
p));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "eqd:= diff(f(t),t,t
)+4*diff(f(t),t)+3*f(t)=g(t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 43 "map(unapply, convert(eqd, D),t); expand(%);" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 59 "map(Lap, %)(p); tmp2:= collect(subs(res81, res82, %
), Lap);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "cond_ini:= f(0)
=0, D(f)(0)=1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "subs(cond
_ini, tmp2); (op@solve)(\{%\}, \{Lap(f)(p)\}):\ntmp3:= collect(%,exp,f
actor);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "tmp4:= subs(lap_
gg, tmp3): collect(%, exp);\ncollect(%, exp, zz->convert(zz,parfrac,p)
);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "fu2:= t->-1/3-1/6*exp
(-3*t)+1/2*exp(-t): vlap(%)(p);\nxplif(%,[p,positive]);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "aa:= 1; fu3:= unapply(fu2(t-aa)*Hea
viside(t-aa),t);\nxlap(fu3)(p); xplif(%,[p,positive]);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "fu1:= t->(1/3-exp(-3*t)/3)*Heavisid
e(t); vlap(%)(p);\n(factor@xplif)(%,[p,positive]);" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 119 "fu:= fu1+fu3; vlap(fu)(p); (xcombipo@expan
d@xplif)(%,[p, positive]);\ncollect(%, exp); (normal@eval@subs)(f=fu, \+
%-tmp4);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "fu(t);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "grb := proc (qqq) options o
perator, arrow; displayf(plot(qqq,t = -1 .. 5,linestyle = 16,color = b
lack),tickmarks = [[0,1,5], 3],axes = boxed) end proc;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "grb(gv(t));" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 11 "grb(fu(t));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "grb(D(fu)(t) );" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "grb((D@@2)(fu)(t) );" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 113 "collect(D(fu)(t), Dirac); xplif(%, [t,real], simplif
y):\nval_dfu:= collect(%, Heaviside, factor@xcombipo@expand);;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "collect(diff(val_dfu,t), Di
rac); xplif(%, [t,real], simplify):\nval_dfu2:= collect(%, Heaviside, \+
factor@xcombipo@expand);;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
47 "collect(val_dfu2+4*val_dfu+3*fu(t), Heaviside);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 107 "eqdg:= (eval@subs)(g=gv, eqd); \ndsol:= \+
(rhs@collect)(dsolve(\{%,f(0)=0, D(f)(0)=1\},f(t)), Heaviside); fu(t);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 142 "eqd_k:= subs(f=k, lhs(
eqd)); dso:= dsolve(%, k(t));\ndef_al:= alpha(t)= subs(dso,_C1=1, _C2=
0, k(t)), beta(t)= subs(dso,_C1=0, _C2=1, k(t));   " }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 98 "def_a:= y(t)=a(t)*alpha(t)+b(t)*beta(t), \+
diff( y(t),t)=a(t)*diff(alpha(t),t)+b(t)*diff(beta(t),t);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "eq1:= subs(def_al, diff(def_a[1],t)
-def_a[2]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "(expand@subs
)(f=y,def_a[2],def_a[1],eqdg);\neq2:= collect((expand@subs)(def_al,%),
[a,b],xcombipo);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "sol:= (
xcombipo@solve)(\{eq1,eq2\},diff(\{a(t),b(t)\},t));" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 163 "a(t)=A+Int(subs(sol,diff(a(t),t)),t); va
:= collect(value(%), Heaviside,xcombipo);\nb(t)=B+Int(subs(sol,diff(b(
t),t)),t); vb:= collect(value(%), Heaviside,xcombipo);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "collect(subs(def_al, va,vb, def_a[1
]), Heaviside, xcombipo@expand);;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 23 "tmp1:= collect(%, Lap);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 62 "coeff(rhs(tmp1),Lap(g)(p)); fv:= unapply(invlaplace(%
,p,t),t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "Int(fv(t-x)*gv
(x),x=0..t); (xcombipo@expand@value)(%): collect(%+fv(t), Heaviside);;
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "dsol;" }}}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 29 "
Autre \351quation diff\351rentielle" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 77 "eqd:= diff(f(t),t,t)+3*diff(f(t),t)+2*f(t)=g(t);\ncon
dini:= f(0)=1, D(f)(0)=2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
91 "(expand@map)(unapply,eqd,t); map(Lap,%)(p);\ntmp:= subs(res81, res
82, (xcombipo@expand)(%));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
56 "(op@solve)(tmp, \{Lap(f)(p)\}): eq_lap0:= collect(%, Lap);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "eq_lap:= collect(subs(condin
i, eq_lap0), Lap, factor);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
95 "L1:= coeff(rhs(eq_lap),Lap(g)(p),0); convert(%, parfrac,p);\nf1:= \+
unapply(invlaplace(%,p,t), t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 94 "L2:= coeff(rhs(eq_lap),Lap(g)(p),1); convert(%, parfrac,p);\nf
2:= unapply(invlaplace(%,p,t),t);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 78 "f1(t) + int( f2(t-x)*sin(x),x=0..t): xcombipo(%); \n(
expand@subs)(f(t)=%, eqd);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0 0" 10 }{VIEWOPTS 1 1 0 1 1 
1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }