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desolve.cc
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desolve.cc
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/* -*- mode:C++ ; compile-command: "g++-3.4 -I.. -g -c desolve.cc -DHAVE_CONFIG_H -DIN_GIAC" -*- */
#include "giacPCH.h"
/*
* Copyright (C) 2000, 2014 B. Parisse, Institut Fourier, 38402 St Martin d'Heres
*
* This program is free software; you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
* the Free Software Foundation; either version 3 of the License, or
* (at your option) any later version.
*
* This program is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU General Public License for more details.
*
* You should have received a copy of the GNU General Public License
* along with this program. If not, see <http://www.gnu.org/licenses/>.
*/
using namespace std;
#include <stdexcept>
#include <cmath>
#include "desolve.h"
#include "derive.h"
#include "intg.h"
#include "subst.h"
#include "usual.h"
#include "symbolic.h"
#include "unary.h"
#include "poly.h"
#include "sym2poly.h" // for equalposcomp
#include "tex.h"
#include "modpoly.h"
#include "series.h"
#include "solve.h"
#include "ifactor.h"
#include "prog.h"
#include "rpn.h"
#include "lin.h"
#include "intgab.h"
#include "giacintl.h"
#include "signalprocessing.h"
#ifndef NO_NAMESPACE_GIAC
namespace giac {
#endif // ndef NO_NAMESPACE_GIAC
gen integrate_without_lnabs(const gen & e,const gen & x,GIAC_CONTEXT){
// workaround for desolve(diff(y)*sin(x)=y*ln(y),x,y);
// otherwise it returns ln(-1-cos(x))
bool save_cv=complex_variables(contextptr);
complex_variables(false,contextptr);
gen res=integrate_gen(e,x,contextptr);
if (lop(res,at_abs).empty() && lop(res,at_floor).empty()){
complex_variables(save_cv,contextptr);
return res;
}
bool save_do_lnabs=do_lnabs(contextptr);
do_lnabs(false,contextptr);
res=integrate_gen(e,x,contextptr);
do_lnabs(save_do_lnabs,contextptr);
complex_variables(save_cv,contextptr);
return res;
}
gen gen_t(const vecteur & v,GIAC_CONTEXT){
#ifdef GIAC_HAS_STO_38
identificateur id_t("t38_");
#else
identificateur id_t(" t");
#endif
gen tmp_t,t=t__IDNT;
t=t._IDNTptr->eval(1,tmp_t,contextptr);
if (t!=t__IDNT || equalposcomp(lidnt(v),t__IDNT))
t=id_t;
return t;
}
gen laplace(const gen & f0,const gen & x,const gen & s,GIAC_CONTEXT){
if (x.type!=_IDNT)
return gensizeerr(contextptr);
if (f0.type==_VECT){
vecteur v=*f0._VECTptr;
for (int i=0;i<int(v.size());++i){
v[i]=laplace(v[i],x,s,contextptr);
}
return gen(v,f0.subtype);
}
gen t(s),pt;
// addition by L.Marohnić: support for transforming periodic functions
if (laplace_periodic(f0,x,s,pt,contextptr))
return pt;
if (s==x){
#ifdef GIAC_HAS_STO_38
t=identificateur("s38_");
#else
t=identificateur(" t");
#endif
}
// check for negative powers of x in f
gen f(f0);
vecteur v(1,x);
lvar(f,v);
fraction ff=sym2r(f,v,contextptr);
gen ffden=ff.den;
int n=0;
if (ffden.type==_POLY){
polynome & ffdenp = *ffden._POLYptr;
if (!ffdenp.coord.empty() && (n=ffdenp.coord.back().index.front()) ){
// multiply by (-1)^n*x^n, do laplace, then integrate n times answer
index_t idxt(v.size());
idxt.front()=-n;
ff=fraction(ff.num,ffden._POLYptr->shift(idxt));
f=r2sym(ff,v,contextptr);
if (n%2)
f=-f;
}
}
if (!assume_t_in_ab(t,plus_inf,plus_inf,true,true,contextptr))
return gensizeerr(contextptr);
int c=calc_mode(contextptr);
calc_mode(0,contextptr);
gen res=_integrate(makesequence(f*exp(-t*x,contextptr),x),contextptr);
calc_mode(c,contextptr);
if (lop(res,at_integrate).empty() && lop(res,at_piecewise).empty() && lop(res,at_sign).empty())
res=-_limit(makesequence(res,x,0,1),contextptr);
else
res=undef;
if (is_undef(res))
res=_integrate(makesequence(f*exp(-t*x,contextptr),x,0,plus_inf),contextptr);
for (int i=1;i<=n;++i){
if (is_undef(res))
return res;
res = _integrate(gen(makevecteur(res,t,0,t),_SEQ__VECT),contextptr);
res += _integrate(gen(makevecteur(f/pow(-x,i),x,0,plus_inf),_SEQ__VECT),contextptr);
}
purgenoassume(t,contextptr);
if (s==x)
res=subst(res,t,x,false,contextptr);
return ratnormal(res,contextptr);
/*
gen remains,res=integrate(f*exp(-t*x,contextptr),*x._IDNTptr,remains,contextptr);
res=subst(-res,x,zero,false,contextptr);
if (s==x)
res=subst(res,t,x,false,contextptr);
if (!is_zero(remains))
res = res +symbolic(at_integrate,gen(makevecteur(remains,x,0,plus_inf),_SEQ__VECT));
return res;
*/
}
static gen _laplace_(const gen & args,GIAC_CONTEXT){
if (args.type!=_VECT)
return laplace(args,vx_var,vx_var,contextptr);
vecteur & v=*args._VECTptr;
int s=int(v.size());
if (s==2)
return laplace( v[0],v[1],v[1],contextptr);
if (s!=3)
return gensizeerr(contextptr);
return laplace( v[0],v[1],v[2],contextptr);
}
// "unary" version
gen _laplace(const gen & args,GIAC_CONTEXT){
if ( args.type==_STRNG && args.subtype==-1) return args;
bool b=approx_mode(contextptr);
approx_mode(false,contextptr);
#if !defined NSPIRE && !defined FXCG
my_ostream * ptr=logptr(contextptr);
logptr(0,contextptr);
gen res=_laplace_(args,contextptr);
logptr(ptr,contextptr);
#else
gen res=_laplace_(exact(args,contextptr),contextptr);
#endif
approx_mode(b,contextptr);
if (b || has_num_coeff(args))
res=simplifier(evalf(res,1,contextptr),contextptr);
return res;
}
static const char _laplace_s []="laplace";
static define_unary_function_eval (__laplace,&_laplace,_laplace_s);
define_unary_function_ptr5( at_laplace ,alias_at_laplace,&__laplace,0,true);
polynome cstcoeff(const polynome & p){
vector< monomial<gen> >::const_iterator it=p.coord.begin(),itend=p.coord.end();
for (;it!=itend;++it){
if (it->index.front()==0)
break;
}
return polynome(p.dim,vector< monomial<gen> >(it,itend));
}
// reduction of a fraction with multiple poles to single poles by integration
// by part, use the relation
// ilaplace(P'/P^(k+1))=laplacevar/k*ilaplace(1/P^k)
pf<gen> laplace_reduce_pf(const pf<gen> & p_cst, tensor<gen> & laplacevar ){
pf<gen> p(p_cst);
assert(p.mult>0);
if (p.mult==1)
return p_cst;
tensor<gen> fprime=p.fact.derivative();
tensor<gen> d(fprime.dim),C(fprime.dim),u(fprime.dim),v(fprime.dim);
egcdpsr(p.fact,fprime,u,v,d); // f*u+f'*v=d
tensor<gen> usave(u),vsave(v);
// int initial_mult=p.mult-1;
while (p.mult>1){
egcdtoabcuv(p.fact,fprime,p.num,u,v,d,C);
p.mult--;
p.den=(p.den/p.fact)*C*gen(p.mult);
p.num=u*gen(p.mult)+v.derivative()+v*laplacevar;
if ( (p.mult % 5)==1) // simplify from time to time
TsimplifybyTlgcd(p.num,p.den);
if (p.mult==1)
break;
u=usave;
v=vsave;
}
return pf<gen>(p);
}
static gen pf_ilaplace(const gen & e0,const gen & x, gen & remains,int,GIAC_CONTEXT){
vecteur vexp;
gen res;
lin(e0,vexp,contextptr); // vexp = coeff, arg of exponential
const_iterateur it=vexp.begin(),itend=vexp.end();
remains=0;
for (;it!=itend;){
gen coeff=*it;
++it;
gen axb=*it,expa,expb;
++it;
gen e=coeff*exp(axb,contextptr);
if (!is_linear_wrt(axb,x,expa,expb,contextptr)){
remains += e;
continue;
}
if (is_strictly_positive(expa,contextptr))
*logptr(contextptr) << gettext("Warning, exponential x coeff is positive ") << expa << '\n';
vecteur varx(lvarx(coeff,x));
int varxs=int(varx.size());
if (!varxs){ // Dirac function
res += coeff*exp(expb,contextptr)*symbolic(at_Dirac,laplace_var+expa);
continue;
}
if ( (varxs>1) || (varx.front()!=x) ) {
remains += e;
continue;
}
vecteur l;
l.push_back(x); // insure x is the main var
l.push_back(laplace_var); // s var as second var
l=vecteur(1,l);
alg_lvar(makevecteur(coeff,axb),l);
gen glap=e2r(laplace_var,l,contextptr);
if (glap.type!=_POLY) return gensizeerr(gettext("desolve.cc/pf_ilaplace"));
int s=int(l.front()._VECTptr->size());
if (!s){
l.erase(l.begin());
s=int(l.front()._VECTptr->size());
}
gen r=e2r(coeff,l,contextptr);
gen r_num,r_den;
fxnd(r,r_num,r_den);
if (r_num.type==_EXT){
remains += e;
continue;
}
if (r_den.type!=_POLY){
remains += e;
continue;
}
polynome den(*r_den._POLYptr),num(s);
if (r_num.type==_POLY)
num=*r_num._POLYptr;
else
num=polynome(r_num,s);
polynome p_content(lgcd(den));
factorization vden(sqff(den/p_content)); // first square-free factorization
vector< pf<gen> > pfde_VECT;
polynome ipnum(s),ipden(s),temp(s),tmp(s);
partfrac(num,den,vden,pfde_VECT,ipnum,ipden);
vector< pf<gen> >::iterator it=pfde_VECT.begin();
vector< pf<gen> >::const_iterator itend=pfde_VECT.end();
vector< pf<gen> > rest,finalde_VECT;
for (;it!=itend;++it){
pf<gen> single(laplace_reduce_pf(*it,*glap._POLYptr));
gen extra_div=1;
factor(single.den,p_content,vden,false,withsqrt(contextptr),complex_mode(contextptr),1,extra_div);
partfrac(single.num,single.den,vden,finalde_VECT,temp,tmp);
}
it=finalde_VECT.begin();
itend=finalde_VECT.end();
gen lnpart(0),deuxaxplusb,sqrtdelta,exppart;
polynome a(s),b(s),c(s);
polynome d(s),E(s),lnpartden(s);
polynome delta(s),atannum(s),alpha(s);
vecteur lprime(l);
if (lprime.front().type!=_VECT) return gensizeerr(gettext("desolve.cc/pf_ilaplace"));
lprime.front()=cdr_VECT(*(lprime.front()._VECTptr));
bool uselog;
for (;it!=itend;++it){
int deg=it->fact.lexsorted_degree();
switch (deg) {
case 1: // 1st order
findde(it->den,a,b);
lnpart=lnpart+rdiv(r2e(it->num,l,contextptr),r2e(firstcoeff(a),lprime,contextptr),contextptr)*exp(r2e(rdiv(-b,a,contextptr),lprime,contextptr)*laplace_var,contextptr);
break;
case 2: // 2nd order
findabcdelta(it->fact,a,b,c,delta);
exppart=exp(r2e(rdiv(-b,gen(2)*a,contextptr),lprime,contextptr)*laplace_var,contextptr);
uselog=is_positive(delta);
alpha=(it->den/it->fact).trunc1()*a;
findde(it->num,d,E);
atannum=a*E*gen(2)-b*d;
// cos part d/alpha*ln(fact)
lnpartden=alpha;
simplify(d,lnpartden);
if (uselog){
sqrtdelta=normal(sqrt(r2e(delta,lprime,contextptr),contextptr),contextptr);
gen racine=ratnormal(sqrtdelta/gen(2)/r2e(a,lprime,contextptr),contextptr);
lnpart=lnpart+rdiv(r2e(d,lprime,contextptr),r2e(lnpartden,lprime,contextptr),contextptr)*cosh(racine*laplace_var,contextptr)*exppart;
gen aa=ratnormal(r2e(atannum,lprime,contextptr)/r2e(alpha,lprime,contextptr)/sqrtdelta,contextptr);
lnpart=lnpart+aa*sinh(racine*laplace_var,contextptr)*exppart;
}
else {
sqrtdelta=normal(sqrt(r2e(-delta,lprime,contextptr),contextptr),contextptr);
gen racine=ratnormal(sqrtdelta/gen(2)/r2e(a,lprime,contextptr),contextptr);
lnpart=lnpart+rdiv(r2e(d,lprime,contextptr),r2e(lnpartden,lprime,contextptr),contextptr)*cos(racine*laplace_var,contextptr)*exppart;
gen aa=ratnormal(r2e(atannum,lprime,contextptr)/r2e(alpha,lprime,contextptr)/sqrtdelta,contextptr);
lnpart=lnpart+aa*sin(racine*laplace_var,contextptr)*exppart;
}
break;
default:
rest.push_back(pf<gen>(it->num,it->den,it->fact,1));
break ;
}
}
vecteur ipnumv=polynome2poly1(ipnum,1);
gen deno=r2e(ipden,l,contextptr);
int nums=int(ipnumv.size());
for (int i=0;i<nums;++i){
gen tmp = rdiv(r2e(ipnumv[i],lprime,contextptr),deno,contextptr);
tmp = tmp*symbolic(at_Dirac,(i==nums-1)?laplace_var:gen(makevecteur(laplace_var,nums-1-i),_SEQ__VECT));
res += tmp;
}
remains += r2sym(rest,l,contextptr)*exp(axb,contextptr);
if (is_zero(expa))
res += lnpart*exp(expb,contextptr);
else
res += quotesubst(lnpart,laplace_var,laplace_var+expa,contextptr)*exp(expb,contextptr)*_Heaviside(laplace_var+expa,contextptr);
}
return res;
}
gen ilaplace(const gen & f,const gen & x,const gen & s,GIAC_CONTEXT){
if (x.type!=_IDNT)
return gensizeerr(contextptr);
if (has_num_coeff(f))
return ilaplace(exact(f,contextptr),x,s,contextptr);
// addition by L.Marohnić: support for periodic summation
gen orig;
if (ilaplace2(f,x,s,orig,contextptr))
return orig;
gen remains,res=linear_apply(f,x,remains,0,contextptr,pf_ilaplace);
res=subst(res,laplace_var,s,false,contextptr);
if (!is_zero(remains))
res=res+symbolic(at_ilaplace,makevecteur(remains,x,s));
return res;
}
// "unary" version
gen _ilaplace(const gen & args,GIAC_CONTEXT){
if ( args.type==_STRNG && args.subtype==-1) return args;
if (args.type!=_VECT)
return ilaplace(args,vx_var,vx_var,contextptr);
vecteur & v=*args._VECTptr;
int s=int(v.size());
if (s==2)
return ilaplace( v[0],v[1],v[1],contextptr);
if (s!=3)
return gensizeerr(contextptr);
return ilaplace( v[0],v[1],v[2],contextptr);
}
static const char _ilaplace_s []="ilaplace";
static define_unary_function_eval (__ilaplace,&_ilaplace,_ilaplace_s);
define_unary_function_ptr5( at_ilaplace ,alias_at_ilaplace,&__ilaplace,0,true);
static const char _invlaplace_s []="invlaplace";
static define_unary_function_eval (__invlaplace,&_ilaplace,_invlaplace_s);
define_unary_function_ptr5( at_invlaplace ,alias_at_invlaplace,&__invlaplace,0,true);
static gen unable_to_solve_diffeq(){
return gensizeerr(gettext("Unable to solve differential equation"));
}
gen diffeq_constante(int i,GIAC_CONTEXT){
#if 0 // def NSPIRE
if (i<5){
const char * tab[]={"o","p","q","r","s"};
return gen(tab[i],contextptr);
}
#endif
#ifdef GIAC_HAS_STO_38
string s("G_"+print_INT_(i));
#else
string s("c_"+print_INT_(i));
#endif
return gen(s,contextptr);
}
// return -1 if f does not depend on y
static int diffeq_order(const gen & f,const gen & y){
vecteur ydepend(rlvarx(f,y));
const_iterateur it=ydepend.begin(),itend=ydepend.end();
// since we did a recursive lvar we dismiss all variables except
// if they begin with derive
int n=-1;
for (;it!=itend;++it){
if (*it==y)
n=giacmax(n,0);
if ( (it->type==_SYMB) && (it->_SYMBptr->sommet==at_derive) ){
gen & g=it->_SYMBptr->feuille;
int m=-1,nder=1;
if ( (g.type==_VECT) && (!g._VECTptr->empty()) ){
m=diffeq_order(g._VECTptr->front(),y);
if (g._VECTptr->size()==3){
gen & gg=g._VECTptr->back();
if (gg.type==_INT_)
nder=gg.val;
}
}
else
m=diffeq_order(g,y);
if (m>=0)
n=giacmax(n,m+nder);
}
}
return n;
}
// true if f is a linear differential equation
// & returns the coefficient in v in descending order
// v has size order+2 with last term=cst coeff of the diff equation
static bool is_linear_diffeq(const gen & f_orig,const gen & x,const gen & y,int order,vecteur & v,int step_info,GIAC_CONTEXT){
v.clear();
gen f(f_orig),a,b,cur_y(y);
gen t=gen_t(makevecteur(x,y,f_orig),contextptr);
for (int i=0;i<=order;++i){
gen ftmp(quotesubst(f,cur_y,t,contextptr));
if (!is_linear_wrt(eval(ftmp,eval_level(contextptr),contextptr),t,a,b,contextptr))
return false;
if (!rlvarx(a,y).empty())
return false;
if (!i)
v.push_back(b);
v.push_back(a);
cur_y=symb_derive(y,x,i+1);
}
reverse(v.begin(),v.end());
if (step_info && v.size()>3)
gprintf("Linear differential equation of coefficients %gen\nsecond member %gen",makevecteur(vecteur(v.begin(),v.end()-1),-v.back()),step_info,contextptr);
return true;
}
static bool find_n_derivatives_function(const gen & f,const gen & x,int & nder,gen & fonction){
if ( (f.type!=_SYMB) || (f._SYMBptr->sommet!=at_derive) ){
nder=0;
fonction=f;
return true;
}
if (f._SYMBptr->feuille.type!=_VECT){
if (!find_n_derivatives_function(f._SYMBptr->feuille,x,nder,fonction))
return false;
++nder;
return true;
}
vecteur & v=*f._SYMBptr->feuille._VECTptr;
if ( (v.size()>1) && (v[1]!=x) )
return false; // setsizeerr(contextptr);
if (!find_n_derivatives_function(v[0],x,nder,fonction))
return false;
if ( (v.size()==3) && (v[2].type==_INT_) )
nder += v[2].val;
else
nder += 1;
return true;
}
static gen function_of(const gen & y_orig,const gen & x_orig){
if ( (y_orig.type!=_SYMB) || (y_orig._SYMBptr->sommet!=at_of) )
return gensizeerr(gettext("function_of"));
vecteur & v =*y_orig._SYMBptr->feuille._VECTptr;
if ( (v[1]!=x_orig) || (v[0].type!=_IDNT) )
return gensizeerr(gettext("function_of"));
return v[0];
}
static gen in_desolve_with_conditions(const vecteur & v_,const gen & x,const gen & y,const gen & solution_generale,const vecteur & parameters,const gen & f,int step_info,GIAC_CONTEXT){
gen yy(y);
vecteur v(v_);
if (yy.type!=_IDNT)
yy=function_of(y,x);
if (is_undef(yy))
return yy;
// special handling for systems
if (solution_generale.type==_VECT && v.size()==2){
gen init=v[1],point=0;
if (init.is_symb_of_sommet(at_equal) && init._SYMBptr->feuille.type==_VECT&& init._SYMBptr->feuille._VECTptr->size()>=2){
point=(*init._SYMBptr->feuille._VECTptr)[0];
init=(*init._SYMBptr->feuille._VECTptr)[1];
if (!point.is_symb_of_sommet(at_of) || point._SYMBptr->feuille.type!=_VECT || point._SYMBptr->feuille._VECTptr->size()<2 || point._SYMBptr->feuille._VECTptr->front()!=y)
return gensizeerr("Bad initial condition");
point=(*point._SYMBptr->feuille._VECTptr)[1];
}
gen systeme=subst(solution_generale,x,point,false,contextptr)-init;
gen s=_solve(makesequence(systeme,parameters),contextptr);
if (s.type!=_VECT)
return gensizeerr("Bad initial condition");
vecteur res;
for (unsigned i=0;i<s._VECTptr->size();++i){
gen tmp=subst(solution_generale,parameters,s[i],false,contextptr);
tmp=ratnormal(tmp,contextptr);
res.push_back(tmp);
}
return res;
}
if (solution_generale.type==_VECT)
*logptr(contextptr) << gettext("Boundary conditions for parametric curve not implemented") << '\n';
// solve boundary conditions
iterateur jt=v.begin()+1,jtend=v.end();
for (unsigned ndiff=0;jt!=jtend;++ndiff,++jt){
if (jt->type==_VECT && jt->_VECTptr->size()==2){
if (ndiff)
*jt=symbolic(at_of,makesequence(symbolic(at_derive,makesequence(y,x,int(ndiff))),jt->_VECTptr->front()))-jt->_VECTptr->back();
else
*jt=symbolic(at_of,makesequence(y,jt->_VECTptr->front()))-jt->_VECTptr->back();
}
}
const_iterateur it=v.begin()+1,itend=v.end();
vecteur conditions(remove_equal(it,itend));
if (conditions.empty())
return solution_generale;
// conditions must be in terms of y(value) or derivatives
vecteur condvar(rlvarx(conditions,yy));
vecteur yvar; // will contain triplet (var,n,x) n=nth derivative, x point
it=condvar.begin(),itend=condvar.end();
int maxnder=0;
for (;it!=itend;++it){
if ( (it->type!=_SYMB) || (it->_SYMBptr->sommet!=at_of) )
continue;
vecteur & w=*it->_SYMBptr->feuille._VECTptr;
int nder;
gen fonction;
if (!find_n_derivatives_function(w[0],x,nder,fonction))
return gensizeerr(contextptr);
if (fonction==y){
if ( (w[1].type==_VECT) && (!w[1]._VECTptr->empty()))
yvar.push_back(makevecteur(*it,nder,w[1]._VECTptr->front()));
else
yvar.push_back(makevecteur(*it,nder,w[1]));
}
if (nder>maxnder)
maxnder=nder;
}
// compute all derivatives of the general solution
vecteur derivatives(1,solution_generale);
gen current=solution_generale;
for (int i=1;i<=maxnder;++i){
current=derive(current,x,contextptr);
derivatives.push_back(current);
}
// evaluate at points of yvar making substition vectors
it=yvar.begin(),itend=yvar.end();
vecteur substin,substout;
for (;it!=itend;++it){
vecteur & w=*it->_VECTptr;
substin.push_back(w[0]);
substout.push_back(subst(derivatives[w[1].val],x,w[2],false,contextptr));
}
// replace in conditions
conditions=*eval(subst(conditions,substin,substout,false,contextptr),eval_level(contextptr),contextptr)._VECTptr;
// solve system over _c0..._cn-1
int save_xcas_mode=xcas_mode(contextptr);
xcas_mode(contextptr)=0;
int save_calc_mode=calc_mode(contextptr);
calc_mode(contextptr)=0;
vecteur parameters_solutions=*_solve(gen(makevecteur(conditions,parameters),_SEQ__VECT),contextptr)._VECTptr;
if (step_info)
gprintf("General solution %gen\nSolving initial conditions\n%gen\nunknowns %gen\nSolutions %gen",makevecteur(solution_generale,conditions,parameters,parameters_solutions),step_info,contextptr);
xcas_mode(contextptr)=save_xcas_mode;
calc_mode(contextptr)=save_calc_mode;
// replace _c0..._cn-1 in solution_generale
it=parameters_solutions.begin(),itend=parameters_solutions.end();
vecteur res;
for (;it!=itend;++it){
gen solgen=eval(subst(solution_generale,parameters,*it,false,contextptr),eval_level(contextptr),contextptr);
// check if f is valid at points where conditions hold (3rd column of yvar)
gen solgenchk=eval(subst(f,y,solgen,false,contextptr),1,contextptr);
bool ok=true;
for (unsigned i=0;i<yvar.size();++i){
gen tmp=subst(solgenchk,x,yvar[i][2],false,contextptr);
if (lidnt(tmp).empty() && !is_zero(simplify(tmp,contextptr))){
ok=false;
break;
}
}
if (ok)
res.push_back(solgen);
}
if (res.size()==1)
return res.front();
return res;
}
static gen desolve_with_conditions(const vecteur & v,const gen & x,const gen & y,gen & f,int step_info,GIAC_CONTEXT){
if (v.empty())
return gensizeerr(contextptr);
int ordre; bool num=false;
vecteur parameters;
gen solution_generale(desolve_f(v.front(),x,y,ordre,parameters,f,step_info,num,contextptr));
if (solution_generale.type!=_VECT) {
gen res= in_desolve_with_conditions(v,x,y,solution_generale,parameters,f,step_info,contextptr);
return num?evalf(res,1,contextptr):res;
}
solution_generale.subtype=0; // otherwise desolve([y'=[[1,2],[2,1]]*y+[x,x+1],y(0)=[1,2]]) fails on the Prime (?)
if (parameters.empty())
return num?evalf(solution_generale,1,contextptr):solution_generale;
iterateur it=solution_generale._VECTptr->begin(),itend=solution_generale._VECTptr->end();
vecteur res;
res.reserve(itend-it);
for (;it!=itend;++it){
if (it->type==_VECT) it->subtype=0;
gen tmp=in_desolve_with_conditions(v,x,y,*it,parameters,f,step_info,contextptr);
if (is_undef(tmp))
return tmp;
if (tmp.type==_VECT)
res=mergevecteur(res,*tmp._VECTptr);
else
res.push_back(tmp);
}
return num?evalf(res,1,contextptr):res;
}
static gen desolve_with_conditions(const vecteur & v,const gen & x,const gen & y,gen & f,GIAC_CONTEXT){
int st=step_infolevel(contextptr);
step_infolevel(0,contextptr);
gen res=desolve_with_conditions(v,x,y,f,st,contextptr);
step_infolevel(st,contextptr);
return res;
}
// f must be a vector obtained using factors
// x, y are 2 idnt
// xfact and yfact should be initialized to 1
// return true if f=xfact*yfact where xfact depends on x and yfact on y only
bool separate_variables(const gen & f,const gen & x,const gen & y,gen & xfact,gen & yfact,int step_info,GIAC_CONTEXT){
const_iterateur jt=f._VECTptr->begin(),jtend=f._VECTptr->end();
for (;jt!=jtend;jt+=2){
vecteur tmp(*_lname(*jt,contextptr)._VECTptr);
if (equalposcomp(tmp,y)){
if (equalposcomp(tmp,x))
return false;
yfact=yfact*pow(*jt,*(jt+1),contextptr);
}
else
xfact=xfact*pow(*jt,*(jt+1),contextptr);
}
if (step_info)
gprintf("Separable variables d%gen/%gen=%gen*d%gen",makevecteur(y,yfact,xfact,x),step_info,contextptr);
return true;
}
bool separate_variables(const gen & f,const gen & x,const gen & y,gen & xfact,gen & yfact,GIAC_CONTEXT){
return separate_variables(f,x,y,xfact,yfact,step_infolevel(contextptr),contextptr);
}
void ggb_varxy(const gen & f_orig,gen & vx,gen & vy,GIAC_CONTEXT){
vecteur lv=lidnt(f_orig);
vx=vx_var;
vy=y__IDNT_e;
#if 0
if (calc_mode(contextptr)==1){
vx=gen("ggbtmpvarx",contextptr);
vy=gen("ggbtmpvary",contextptr);
}
#endif
for (unsigned i=0;i<lv.size();++i){
string s=lv[i].print(contextptr);
char c=s[s.size()-1];
if (c=='x')
vx=lv[i];
if (c=='y')
vy=lv[i];
}
}
static gen desolve_cleanup(const gen & i,const gen & x,GIAC_CONTEXT){
if (i.is_symb_of_sommet(at_prod)){
gen f=i._SYMBptr->feuille;
if (f.type==_VECT){
vecteur w;
for (int j=0;j<f._VECTptr->size();++j){
gen tmp=desolve_cleanup((*f._VECTptr)[j],x,contextptr);
if (!is_one(tmp))
w.push_back(tmp);
}
return _prod(w,contextptr);
}
}
if (i.is_symb_of_sommet(at_abs) || i.is_symb_of_sommet(at_neg))
return desolve_cleanup(i._SYMBptr->feuille,x,contextptr);
if (is_zero(derive(i,x,contextptr)))
return 1;
return i;
}
// solve linear diff eq of order 1 a*y'+b*y+c=0
static gen desolve_lin1(const gen &a,const gen &b,const gen & c,const gen & x,vecteur & parameters,int step_info,GIAC_CONTEXT){
if (step_info)
gprintf("Linear differential equation of order 1 a*y'+b*y+c=0\na=%gen, b=%gen, c=%gen",makevecteur(a,b,c),step_info,contextptr);
if (a.type==_VECT){
// y'+inv(a)*b(x)*y+inv(a)*c(x)=0
// take laplace transform
// p*Y-Y(0)+bsura*Y+csura=0
// (p+bsura)*Y=Y(0)-csura
int n=int(a._VECTptr->size());
if (!ckmatrix(a) || !ckmatrix(b))
return gensizeerr(contextptr);
gen inva=inv(a,contextptr);
gen bsura=inva*b,csura,cl;
if (!is_zero(derive(bsura,x,contextptr)))
return gensizeerr("Non constant linear differential system");
if (c.type==_VECT){
vecteur & cv=*c._VECTptr;
for (unsigned i=0;i<cv.size();++i){
if (cv[i].type==_VECT && cv[i]._VECTptr->size()==1)
cv[i]=cv[i]._VECTptr->front();
}
csura=inva*c;
cl=_laplace(makesequence(csura,x,x),contextptr);
}
else {
if (!is_zero(c))
return gensizeerr("Invalid second member");
cl=vecteur(n);
}
if (cl.type!=_VECT || int(cl._VECTptr->size())!=n)
return gensizeerr("Invalid second member");
for (int i=0;i<n;++i){
parameters.push_back(diffeq_constante(int(parameters.size()),contextptr));
(*cl._VECTptr)[i] = parameters.back()- (*cl._VECTptr)[i];
}
cl=inv(bsura+x,contextptr)*cl;
cl=ilaplace(cl,x,x,contextptr);
return vecteur(1,ratnormal(cl,contextptr));
}
gen i=integrate_without_lnabs(inv(a,contextptr)*b,x,contextptr);
i=normal(lnexpand(i,contextptr),contextptr);
i=exp(i,contextptr);
if (step_info)
gprintf("Homogeneous solution C/%gen",makevecteur(i),step_info,contextptr);
i=expexpand(i,contextptr);
i=simplify(i,contextptr);
// cleanup general solution: remove cst factors and absolute values
i=desolve_cleanup(i,x,contextptr);
gen C=integrate_without_lnabs(ratnormal(rdiv(-c,a,contextptr)*i,contextptr),x,contextptr);
if (step_info && C!=0)
gprintf("Particuliar solution %gen",makevecteur(C),step_info,contextptr);
parameters.push_back(diffeq_constante(int(parameters.size()),contextptr));
gen res=ratnormal(_lin((C+parameters.back())/i,contextptr),contextptr);
if (step_info)
gprintf("General solution %gen",makevecteur(res),step_info,contextptr);
return res;
}
bool desolve_linn(const gen & x,const gen & y,const gen & t,int n,vecteur & v,vecteur & parameters,gen & result,int step_info,GIAC_CONTEXT){
// 1st order
if (n==1){ // a(x)*y'+b(x)*y+c(x)=0
// y'/y=-b/a -> y=C(x)exp(-int(b/a)) and a(x)*C'*exp()+c(x)=0
gen & a=v[0];
gen & b=v[1];
gen & c=v[2];
if (ckmatrix(a)){
if (c.type!=_VECT && is_zero(c))
c=c*a;
c=_tran(c,contextptr)[int(a._VECTptr->size())-1];
}
result=desolve_lin1(a,b,c,x,parameters,step_info,contextptr);
return true;
}
// cst coeff?
gen cst=v.back();
v.pop_back();
if (derive(v,x,contextptr)==vecteur(n+1,zero)){
if (step_info)
gprintf("Linear differential equation with constant coefficients\nOrder %gen, coefficients %gen",makevecteur(n,v),step_info,contextptr);
// Yes!
// simpler general solution for small order generic lin diffeq with cst coeff/squarefree case
if (n<=3){
vecteur rac=solve(horner(v,x,contextptr),x,1,contextptr);
comprim(rac);
if (n==2 && rac.size()==1){
parameters.push_back(diffeq_constante(int(parameters.size()),contextptr));
parameters.push_back(diffeq_constante(int(parameters.size()),contextptr));
gen sol = exp(rac.front()*x,contextptr)*(parameters[parameters.size()-2]*x+parameters.back());
if (step_info)
gprintf("Homogeneous solution %gen",makevecteur(sol),step_info,contextptr);
bool b=calc_mode(contextptr)==1;
if (b)
calc_mode(0,contextptr);
gen part=_integrate(makesequence(-cst/v.front()*exp(-rac.front()*x,contextptr),x),contextptr)*x+_integrate(makesequence(cst/v.front()*x*exp(-rac.front()*x,contextptr),x),contextptr);
if (step_info)
gprintf("Particuliar solution %gen",makevecteur(part),step_info,contextptr);
if (b)
calc_mode(1,contextptr);
part=simplify(part*exp(rac.front()*x,contextptr),contextptr);
result=sol+part;
if (step_info)
gprintf("General solution %gen",makevecteur(result),step_info,contextptr);
return true;
}
if (int(rac.size())==n){
gen sol; bool reel=true;
for (int j=0;j<n;){
if (j<n-1 && is_zero(ratnormal(rac[j]-conj(rac[j+1],contextptr),contextptr),contextptr)){
gen racr,raci;
reim(rac[j],racr,raci,contextptr);
if (is_strictly_positive(-raci,contextptr))
raci=-raci;
parameters.push_back(diffeq_constante(int(parameters.size()),contextptr));
parameters.push_back(diffeq_constante(int(parameters.size()),contextptr));
sol += exp(racr*x,contextptr)*(parameters[parameters.size()-2]*cos(raci*x,contextptr)+parameters[parameters.size()-1]*sin(raci*x,contextptr));
j+=2;
continue;
}
if (reel && !is_zero(im(rac[j],contextptr)))
reel=false;
parameters.push_back(diffeq_constante(int(parameters.size()),contextptr));
sol += parameters.back()*exp(rac[j]*x,contextptr);
j++;
}
if (step_info)
gprintf("Homogeneous solution %gen",makevecteur(sol),step_info,contextptr);
if (derive(cst,x,contextptr)==0 && !is_zero(v.back())){
result=sol-cst/v.back();
return true;
}
// variation des constantes
gen M_=_vandermonde(rac,contextptr),part=0;
if (ckmatrix(M_)){
matrice M=*M_._VECTptr;
vecteur c(n);
c[n-1]=-_trig2exp(cst,contextptr)/v.front();
c=linsolve(mtran(M),c,contextptr);
for (unsigned i=0;i<c.size();++i){
bool b=calc_mode(contextptr)==1;
if (b)
calc_mode(0,contextptr);
gen tmp=_lin(makesequence(c[i]*exp(-rac[i]*x,contextptr),at_sqrt),contextptr);
tmp = _integrate(makesequence(tmp,x),contextptr);
part += _lin(makesequence(tmp*exp(rac[i]*x,contextptr),at_sqrt),contextptr);
if (b)
calc_mode(1,contextptr);
}
if (reel && is_zero(im(cst,contextptr)) && lop(part,at_integrate).empty())
part=re(part,contextptr);
if (1) // desolve((y''+y=sin(x)) and (y(0)=1) and (y'(0)=2),y)
part=recursive_ratnormal(part,contextptr);
else
part=simplify(part,contextptr);
}
if (step_info)
gprintf("Particuliar solution %gen",makevecteur(part),step_info,contextptr);
result=sol+part;
return true;
}
} // end n<=3
gen laplace_cst=_laplace(makesequence(-cst,x,t),contextptr);
if (!is_undef(laplace_cst)){
vecteur lopei=mergevecteur(lop(laplace_cst,at_Ei),lop(laplace_cst,at_integrate));
if (lopei.empty()){
gen arbitrary,tmp;
for (int i=n-1;i>=0;--i){
parameters.push_back(diffeq_constante(int(parameters.size()),contextptr));
tmp=tmp*t+parameters.back();
arbitrary=arbitrary+v[i]*tmp;
}
arbitrary=(laplace_cst+arbitrary)/symb_horner(v,t);
arbitrary=ilaplace(arbitrary,t,x,contextptr);
result=arbitrary;
return true;
}
}
}
if (n==2){ // a(x)*y''+b(x)*y'+c(x)*y+d(x)=0
gen & a=v[0];
gen & b=v[1];
gen & c=v[2];
gen & d=cst;
#if 0
if (is_exactly_zero(c)){
vecteur v1(makevecteur(a,b,d));
if (desolve_linn(x,y,t,1,v1,parameters,result,step_info,contextptr)){
result=_integrate(makesequence(result,x),contextptr);
return true;
}
}
#endif
gen u=-b/a,V=-c/a,w=-d/a,
k=simplify(u*u/4-derive(u,x,contextptr)/2+V,contextptr);
// y''=u*y'+V*y+w (with u,V,w functions of x)
// Pseudo-code from fhub on HP Museum Forum
/*
k:=u^2/4-u'/2+V
if k==const or k*x^2=const then
if k=const
then s:=x; t:=e^(int(u,x)/2);
else u:=u*x+1; k:=u^2/4+V*x^2; s:=ln(x); t:=x^(u/2);
endif;
if k=0 then u:=t*s; V:=t;
elseif k>0 then u:=t*e^(sqrt(k)*s); V:=t*e^(-sqrt(k)*s);
else u:=t*cos(sqrt(-k)*s); V:=t*sin(sqrt(-k)*s);
endif;
w:=w/(u*V'-V*u'); w:=V*int(u*w,x)-u*int(V*w,x);
solution: y=c1*u+c2*V+w
endif
*/
bool cst=is_zero(derive(k,x,contextptr));
bool x2=is_zero(derive(ratnormal(u*x,contextptr),x,contextptr)) && is_zero(derive(ratnormal(v*x*x,contextptr),x,contextptr));
if (cst || x2){
gen s,t;
if (cst){
s=x;
t=simplify(exp(integrate_without_lnabs(u,x,contextptr)/2,contextptr),contextptr);
}
else {
u=u*x+1;
u=simplify(u,contextptr);
k=simplify(u*u/4+V*x*x,contextptr);
s=ln(x,contextptr); t=pow(x,u/2,contextptr);
}
if (is_zero(k)){
u=t*s; V=t;
}
else {
if (is_strictly_positive(-k,contextptr)){
gen tmp=sqrt(-k,contextptr)*s;
u=t*cos(tmp,contextptr);
V=t*sin(tmp,contextptr);
}
else {
if (s.is_symb_of_sommet(at_ln)){
gen tmp=pow(s._SYMBptr->feuille,sqrt(k,contextptr),contextptr);
u=t*tmp;
V=t/tmp;
}
else {
gen tmp=sqrt(k,contextptr)*s;
u=t*exp(tmp,contextptr);
V=t*exp(-tmp,contextptr);
}
}
}
w=simplify(w/(u*derive(V,x,contextptr)-V*derive(u,x,contextptr)),contextptr);
w=V*integrate_without_lnabs(u*w,x,contextptr)-
u*integrate_without_lnabs(V*w,x,contextptr);
parameters.push_back(diffeq_constante(int(parameters.size()),contextptr));
parameters.push_back(diffeq_constante(int(parameters.size()),contextptr));
result=w+parameters[parameters.size()-2]*u+parameters[parameters.size()-1]*V;
return true;
}
// IMPROVE: if a, b, c are polynomials, search for a polynomial solution
// of the homogeneous equation, if found we can solve the diffeq
gen aa(a),bb(b),cc(c);
if (lvarxwithinv(makevecteur(a,b,c),x,contextptr)==vecteur(1,x)){
vecteur l=vecteur(1,x);
gen a0(a),b0(b);
a=_coeff(makesequence(a,x),contextptr);
b=_coeff(makesequence(b,x),contextptr);
c=_coeff(makesequence(c,x),contextptr);
if (a.type==_VECT && b.type==_VECT && c.type==_VECT){
int A=int(a._VECTptr->size())-1,B=int(b._VECTptr->size())-1,C=int(c._VECTptr->size())-1,N=-1;
if (C==B-1){
gen n=-c._VECTptr->front()/b._VECTptr->front();
if (n.type==_INT_ && n.val>N){
if (A-2<C || n==1)
N=n.val;
}
if (A-2==C){
// a*n*(n-1)+b*n+c=a*n^2+(b-1)*n+c=0
gen aa=a._VECTptr->front(),bb=b._VECTptr->front()-1,cc=c._VECTptr->front();
gen delta=(sqrt(bb*bb-4*aa*cc,contextptr)+bb)/2;
if (delta.type==_INT_ && delta.val>N)
N=delta.val;
}
}
if (A-2==B-1 && C<B-1){
gen n=-b._VECTptr->front()/a._VECTptr->front()+1;
if (n.type==_INT_ && n.val>N)
N=n.val;
}