
/*
odes  package for solving ordinary diferential equations
version 2.01,  2013.11
Copyright (C)  A.Domarkas 2013
odes package is free software; you can redistribute it and/or
modify it under the terms of the GNU General Public License, 
*/

/*
Examples see in odes-doc.pdf
*/

/*
1
*/

odecv(tr,eq,y,x):=block([_n,_t,tr1,itr,eq1,r,solvetrigwarn],
solvetrigwarn:false,_n:derivdegree(eq,y,x),
tr1:(solve(tr,x))[1],
_t:last(listofvars(rhs(tr1))),
itr:(solve(tr,_t))[1],
depends(y,_t,_t,x),
eq1:ev(eq,nouns),
makelist(diff(itr,x,k),k,1,_n),
subst(%%,eq1),
ev(%%,nouns),
r:subst(tr1,%%),
remove([y,_t],dependency),
radcan(r))$

/*
2
*/

dchange(tr,eq,fun,var,nfun,nvar):=block(local(T,keit),
diffpf(f,t,k):=block(
if k = 0 then f[2] elseif k = 1
then ratsimp(diff(f[2],t,1)/diff(f[1],t,1))
else ratsimp(diff(diffpf(f,t,k-1),t,1)/diff(f[1],t,1))),
T:makelist('diff(fun,var,k)=
diffpf([rhs(tr),nfun],nvar,k),k,0,derivdegree(eq,fun,var)),
keit:append(reverse(T),[tr]),
ev(eq,keit))$

/*
3
*/

odeC(eq,r,x):=block([derivsubst,_u,tr,itr,L],
depends(_u,x),assume(_u>0),
derivsubst:true,
L:listofvars(r),
tr:_u=r, itr:solve(tr,last(L))[1],
subst(itr,eq),
ev(%%, nouns), expand(%%),
ode2(%%,_u,x),
subst(tr,expand(%%))
)$

/*
4
*/

solvet(eq,x):=block([spr,k,maperror,mapprint],
maperror:false,mapprint:false,
spr:solve(eq,x),
rectform(%%),
if freeof(sin,%%) then return(sort(%%)) 
else makelist(x=map(polarform,rhs(spr[k])),k,1,length(spr)),
rectform(%%),
trigsimp(%%),
sort(%%))$

/*
5
*/

ode1_ic(eq,y,x,_ic):=block([sol,gsol, spr,ats,p],
solvetrigwarn:false,
sol:ode2(eq,y,x),
if %%=false then return(false),
trigsimp(sol),
radcan(%%),
trigsimp(%%),
trigreduce(%%), 
trigexpand(%%),
gsol:rootscontract(%%),
subst([x=_ic[1],y=_ic[2]],trigsimp(gsol)),
solve(%%,%c),
subst(%%,gsol),
sol:ratsimp(logcontract(%%)),
spr:solve(%%,y), 
if not freeof(y,map(rhs,%%)) then return(sol),
if length(spr)=1 and freeof(y,rhs(spr[1])) then return(spr[1]),
sublist(spr,lambda([e],is(radcan(subst([x=_ic[1], y=_ic[2]],e))))),
trigsimp(%%[1]),
radcan(%%),
if not freeof(%e,%%) then
radcan(%%) else %%,
if not freeof(%i,%%) then
ats:rootscontract(subst(%i=sqrt(-1),%%))
else ats:%%,
return(ats)
)$

/*
6
*/

ode2_ic(eq,y,x,_ic):=block([p,sol,ats,spr],
solvetrigwarn:false,
maperror:false,
mapprint:false,
assume(x>_ic[1]),
sol:ode2(eq,y,x),
solve(eq,'diff(y,x,2)),
rhs(%%[1]),
if listofvars(%%)=[y] and listp(sol) then
         (
          subst('diff(y,x,2)=p*'diff(p,y),eq),
          ode1_ic(%%,p,y,[_ic[2],_ic[3]]),
          subst(p='diff(y,x),%%),
          ats:ode1_ic(%%,y,x,[_ic[1],_ic[2]])
         )
else
        (
         ic2(sol, x=_ic[1], y=_ic[2],'diff(y,x)=_ic[3]),
         ats:lhs(%%)=map(factor,rhs(%%)),
         if lhs(%%)#y then
              (
               spr:solve(ats,y),
               if spr#[] then
                    (
                     sublist(spr,lambda([e],
                     _ic[2]=ratsimp(ev(rhs(e),x=_ic[1])) and
                     _ic[3]=ratsimp(at(diff(rhs(e),x),x=_ic[1])))),
                     ats:%%[1]
                    )
              ),
        forget(facts(x)),
        return(ats)
        )
)$

/*
6a
*/

ode_ic(eq,y,x,_ic):=block([],
if  derivdegree(eq,y,x)=1 then ode1_ic(eq,y,x,_ic)
elseif derivdegree(eq,y,x)=2 then ode2_ic(eq,y,x,_ic)
else error("equation must be the first or second order"),
rootscontract(%%)
)$

/*
7
*/

P_iter(eq,x,y,x0,y0,n):=block([f,t],
solve(eq,'diff(y,x,1))[1],
define(f(x,y),rhs(%%)),
if n=0 then y0 else 
(assume(x>x0),
subst(x=t,f(t,P_iter(eq,x,y,x0,y0,n-1))),
y0+expand(integrate(%%,t,x0,x))))$

/*
8
*/

ode1taylor(eq,x0,y0,n):=block([spr,nezin],
atvalue(y(x),x=x0,y0),
spr:taylor(y(x),x,x0,n),
nezin:at(makelist(diff(y(x),x,k),k,1,n),x=x0),
makelist(diff(eq,x,k),k,0,n-1),
at(%%,x=x0),solve(%%,nezin),
ratexpand(ev(spr,first(%%))),
taylor(%%,x,x0,n)
)$

/*
9
*/

ode2taylor(eq,x0,y0,y1,n):=block([spr,nezin],
atvalue(y(x),x=x0,y0),
atvalue(diff(y(x),x),x=x0,y1),
spr:taylor(y(x),x,x0,n),
nezin:at(makelist(diff(y(x),x,k),k,2,n),x=x0),
makelist(diff(eq,x,k),k,0,n-2),
at(%%,x=x0),solve(%%,nezin),
ratexpand(ev(spr,first(%%))),
taylor(%%,x,x0,n)
)$

/*
10
*/

ode1exact(eq):=block([P,Q,F],
P:coeff(expand(lhs(eq)-rhs(eq)),dx),
Q:coeff(expand(lhs(eq)-rhs(eq)),dy),
if trigsimp(trigexpand(diff(P,y)-diff(Q,x)))=0 then
(F:integrate(P,x)+g(y),diff(F,y)=Q,
ode2(%%,g(y),y),
subst([%%,%c=0],F)=C,
trigexpand(%%),
trigsimp(%%),
expand(%%))
else false)$

/*
11
*/

intfactor1(eq,omega):=block([P,Q,d,t,ans],
P:coeff(expand(lhs(eq)-rhs(eq)),dx),
Q:coeff(expand(lhs(eq)-rhs(eq)),dy),
(diff(P,y)-diff(Q,x))/(Q*diff(omega,x)-P*diff(omega,y)),
ratsimp(ev(%%,diff)),
trigexpand(%%),
ratsubst(t,omega,%%),
d:map(radcan,expand(%%)),
if listofvars(%%)=[t] or listofvars(%%)=[] then (
exp(integrate(d,t)),
ans:subst(t=omega,%%))
else return(false),
if freeof(integrate, ans) then return(ans)
else return(false)
)$

/*
12
*/

odeL(eq,y,x):=block([n,_yp,gamma_expand,i,_Y,_C],
gamma_expand:true,
_Y:fs(eq,y,x),
n:derivdegree(eq,y,x),
_yp:partsol(eq,y,x),
_C:makelist(concat(C,i),i,1,n),
y=_C._Y+_yp)$

/*
13
*/

odeL_ic(eq,y,x,ic):=block(odeL(eq,y,x), icn(%%,y,x,ic))$

icn(sol,y,x,ic):=block([_spr,n],
_spr:rhs(sol),
n:length(ic)-1,
makelist(at(diff(_spr,x,k),x=ic[1])=ic[k+2],k,0,n-1),
solve(%%,_C:makelist(concat(C,i),i,1,n)),
y=subst(%%,_spr)
)$

/*
14
*/

fs(eq,y,x):=block([_f,n,j,cr,_k,ce],
_f(r):=block([a,b,i],
       a:radcan(realpart(r[1])),
       b:radcan(imagpart(r[1])),
       if b=0 then 
       return(makelist(x^(i-1)*exp(a*x),i,1,r[2]))
       else
       join(makelist(x^(i-1)*exp(a*x)*cos(b*x),i,1,r[2]),
       makelist(x^(i-1)*exp(a*x)*sin(b*x),i,1,r[2]))),
n:derivdegree(eq,y,x),
makelist('diff(y,x,j)=_k^j,j,0,n),
reverse(%%),
ce:subst(%%,lhs(eq))=0,
solve(ce,_k),
map(rhs,%%),
map(rootssimp,%%),
makelist([%%[j],multiplicities[j]],j,1,length(%%)),
cr:sublist(%%,lambda([e],is(radcan(imagpart(e[1]))>=0))),
map(_f,cr),
flatten(%%),
trigreduce(%%),
sort(%%))$

/*
15
*/

partsolL_UC(eq,y,x):=block([spr,u],
if rhs(eq)=0 then return(0),
subst(y=u(x),eq),
rhs(desolve([%%],[u(x)])),
args(%%),
sublist(%%,lambda([e],freeof(u,ilt,e))),
spr:apply("+",%%),
if not atom(spr) and op(spr)="+" then
(args(expand(spr)),
sublist(%%,lambda([e],ode_check(lhs(eq),y=e)#0) ),
apply("+",%%),return(ratsimp(%%)))
else
return(ratsimp(spr))
)$

partsolL_VP(eq,y,x):=block([W,Wi,_n,_dp],
if rhs(eq)=0 then return(0),
FS:fs(eq,y,x),
F:rhs(eq),
W:wronskian(FS,x),
_n:length(FS),
makelist(0,i,1,_n-1),
_dp:append(%%,[F]),
Wi:radcan(trigsimp(invert(W))),
integrate(Wi._dp,x),
ratsimp(%%),
trigsimp(%%),
logcontract(%%),
FS.%%,
ratsimp(%%),
trigsimp(%%),
trigreduce(%%),
ratsimp(%%)
)$

partsol(eq,y,x):=block([s,eq1],
eq1:subst(y=y(x),eq),
if freeof(laplace,laplace(rhs(eq),x,s))
and freeof(log,tan,atan,sec,csc,rhs(eq))
then partsolL_UC(eq,y,x) 
else partsolL_VP(eq,y,x)
)$

/*
16
*/

odeM(A,F,t):=block([W,gamma_expand,n,Ya],
gamma_expand:true,
n:length(F),
W:matrix_exp(A,t),
W.integrate(trigsimp(invert(W).F),t),
trigrat(expand(%%)),
Ya:expand(%%),
W.transpose(makelist(concat(C,i),i,1,n))+Ya)$

/*
17
*/

odeM_ic(A,B,t,t0,Y0):=block([bspr],
bspr:odeM(A,B,t),
subst(t=t0,%%)-Y0,
list_matrix_entries(%%),
solve(%%,makelist(concat(C,i),i,1,length(Y0))),
subst(%%[1],bspr),
expand(%%)
)$

/*
18
*/

matrix_exp(A,r):=block([n,B,s,t,Lap,f],
n:length(A),
B:invert(s*ident(n)-A),
Lap(f):=ilt(f, s, t),
matrixmap(Lap,B),
subst(t=r,%%))$

/*
19
*/

odelinsys(A,F,t,t0,Y0):=block([s,g,expA],
assume(t>t0),
define(g(t),F),
mat_function(exp,A*t),ratsimp(%%),
define(Aexp(t),%%),
Aexp(t-t0).Y0+integrate(Aexp(t-s).g(s),s,t0,t),
expand(%%)
)$

/*
20
*/

wronskian(functlist,var):=block([end],end:length(functlist)-1,functlist:[
functlist],thru end do functlist:
endcons(map(lambda([x],diff(x,var)),last(functlist)),functlist),
apply('matrix,functlist))$

/*
etc
*/

itr2can(eq,y,x):=block(coeff(lhs(eq),'diff(y,x))/coeff(lhs(eq),'diff(y,x,2)),
t=radcan(integrate(exp(-integrate(%%,x)),x)))$

rootssimp(x):=block([maperror,mapprint],
maperror:false,
mapprint:false, 
map(polarform,x),rectform(%%),trigsimp(%%))$

trigvalue(r):=block(
[f,x,sol,spr,spr1,solvetrigwarn,algebraic],
solvetrigwarn:false,
algebraic:true,
if freeof(%pi,r) then return(r),
f:part(r,0),
if part(r,0)="-" then f:part(r,1,0),
if f=cot then f:tan,
sol:solve(x=r,%pi)[1],
sol*denom(rhs(sol)),
map(f,%%),
trigexpand(%%),
factor(%%),
spr:solve(%%,x),
if (length(spr)<=2 or not freeof(%i,%%)) then return(r),
spr1:sublist(spr,lambda([e],is(abs(rhs(e)-r)<ratepsilon))),
if %%=[] then return(r),
rhs(spr1[1]),
sqrtdenest(%%),
factor(%%)
)$

