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SETUP_AUTOLOAD("lodsav",LODESAVE)$
svhypr(wb,wc):=block([p],swsv:t,print("the type is hypergeometric"),
for i:1 thru 3 do (sl[i]:ssolve(cheq[i],l), la[i,0]:spt[i],
la[i,1]:part(sl[i],1,2), la[i,2]:part(sl[i],2,2)),
print("the solution may be written by Riemann's P-functions as follows"),
pfuncthg(n,dum), replcompform(dum), infosum(dum),
lmin:min(l[1],l[2],l[3],l[4]), lmax:max(l[1],l[2],l[3],l[4]),
if lmin=0 then (j:seekodd(dum),
if j>0 then (ldf3(1),ys:nopmt(j),return(ys))),
ldf3(2),ys:haspmts(dum),return(ys))$
pfuncthg(selv,v):=block([p], p:matrix([la[1,0],la[2,0],la[3,0]],
[la[1,1],la[2,1],la[3,1]],[la[1,2],la[2,2],la[3,2]]),
if selv=y then print("y=",v,"P",p,"(x)") else print("y=P",p,"(x)"))$
infosum(dum):=block([ws],
for i:1 thru 3 do dla[i]:la[i,1]-la[i,2],
ws:dla[1]+dla[2]+dla[3], sum[1]:ratsimp(ws),
ws:dla[1]+dla[2]-dla[3], sum[2]:ratsimp(ws),
ws:dla[1]-dla[2]+dla[3], sum[3]:ratsimp(ws),
ws:dla[1]-dla[2]-dla[3], sum[4]:ratsimp(ws),
for i:1 thru 4 do (var[i]:listofvars(sum[i]),l[i]:length(var[i])))$
excrow(dum):=block([wla], for i:1 thru 3 do exc1c(i))$
ssolve(e,x):=block([l,ws,fws],ws:solve(e,x),l:length(ws),
if l=1 then (fws:first(ws),ws:[fws,fws]),return(ws))$
oddintp(pol,v):=block([c,wtst,wd],c:coeff(pol,v,0),wtst:ratsimp((pol-c)/2),
wd:denom(wtst),if wd=1 and oddp(c) then return(t) else return(f))$
exc1c(i):=block([wla], wla:la[i,1],la[i,1]:la[i,2],
la[i,2]:wla, dla[i]:-dla[i])$
seekodd(dum):=block([i,j],i:1,j:0,
loop, if l[i]=0 and oddp(sum[i]) then (j:j+1,jodd[j]:i),
if l[i]=1 then (ans:read("is",sum[i],"an odd integer?"),
if ans=y then (j:j+1,jodd[j]:i)),
if i<4 then (i:i+1,go(loop)), return(j))$
replcompform(dum):=block([],for i:1 thru 3 do
(nv:listofvars(la[i,1]), if length(nv)>1 then
(ans:read("do you replace in",la[i,1],"?",str4),
if ans=y then (ob:read("replace"),oa:read("by"),
for j:1 thru 2 do la[i,j]:subst(oa,ob,la[i,j])))))$
ldf3(i):=block([], if ldsw[3,i]#y then (ldsw[3,i]:y,
if i=1 then load(pnopm) else
if i=2 then load(phghp) else
if i=3 then load(psleg) else
if i=4 then load(phyp25) else
if i=5 then load(psolh) ))$
LODESave([phypgm,fasl],svhypr,infosum,excrow,ssolve,pfuncthg,
oddintp,exc1c,seekodd,replcompform,ldf3);
/* the sum has no parameter */
nopmt(j):=block([], iodd:minabssum(j),excroots(iodd),
if integerp(sum[iodd]) then (if sum[iodd]>0 then
(excrow(dum),sum[iodd]:-sum[iodd]))
else (ans:read("is",sum[iodd],"positive?",str4), if ans=y then
(excrow(dum),sum[iodd]:-sum[iodd])),
wsum:sum[iodd],xa:x-spt[1],xb:x-spt[2],
print(str1), pfuncthg(n,v), wd:xa*xb,
if sum[iodd]=-1 then ys:elmrephg1(iodd)
else ys:elmrephg2((1-wsum)/2), return(ys))$
minabssum(j):=block([i,j1,j2,minj],i:1,j1:jodd[i], minj:j1,
loop,if i<j then (i:i+1,j2:jodd[i],
if abs(sum[j1]) > abs(sum[j2]) then minj:j2, go(loop)),
return(minj))$
excroots(iodd):=block([],
if iodd=2 or iodd=4 then exc1c(3),
if iodd=3 or iodd=4 then exc1c(2))$
elmrephg1(iodd):=block([w1,w2,w3,w4],
sdla1:ratsimp(-dla[1]-1), sdmu1:ratsimp(-dla[2]-1),
if la[1,1]=la[2,1] then w1:wd^la[1,1]
else w1:xa^la[1,1]*xb^la[2,1],
if dla[1]=dla[2] then w2:wd^sdla1 else w2:xa^sdla1*xb^sdmu1,
w3:k1+k2*'integrate(w2,x), w4:w1*w3, print("y=",w4), ys:w4,
reqans1(dum), if ans=y then
(w5:k1+k2*integrate(w2,x), ys:expand(w1*w5),
y1:coeff(ys,k1,1), y2:coeff(ys,k2,1)), return(ys))$
elmrephg2(m):=block([w1,w2,w3,w4,w5],
sdlam :ratsimp(dla[1]+m), sdmum :ratsimp(dla[2]+m),
sdlam1:ratsimp(sdlam-1),sdmum1:ratsimp(sdmum-1),
if la[1,2]=la[2,2] then w1:wd^la[1,2]
else w1:xa^la[1,2]*xb^la[2,2],
if dla[1]=dla[2] then (w2:wd^sdlam1, w3:wd^(-sdlam)) else
(w2:xa^sdlam1*xb^sdmum1,w3:xa^(-sdlam)*xb^(-sdmum)),
w4:k1+k2*'integrate(w3,x), w5:w1*'diff(w2*w4,x,m-1), print("y=",w5),
ys:w5, reqans1(dum), if ans=y then
(w6:k1+k2*'integrate(w3,x), w7:w1* diff(w2*w6,x,m-1), ys:expand(w7),
y1:coeff(ys,k1,1), y2:coeff(ys,k2,1)), return(ys) )$
LODESave([pnopm,fasl],nopmt,minabssum,excroots,elmrephg1,elmrephg2);
/* the sum has parameters */
haspmts(dum):=block([], y0:1, plist:listofpars(eq),
for i:1 thru 3 do checkcontratn(i),
sortratn(dum), exccolumns(dum), pfuncthg(n,dum),
if not(cdla[1]=cdla[2]) then hgstdd(dum),
make01p(1), make01p(2), pfuncthg(y,y0), xa:x-spt[1],xb:x-spt[2],
if la[1,2]=0 and la[2,2]=0 then (excrow(dum), pfuncthg(y,y0)),
if la[1,1]=0 and la[2,1]=0 and dla[1]=dla[2] then (ans:n,
if not integerp(dla[1]) then
ans:read("Is",dla[1],"an integer?",str4),
if integerp(dla[1]) or ans=y then
(v:-la[3,1],ldf3(3),gsvlgdre(v),return(ys))),
if dla[1]=dla[2] and la[1,3]=2 and la[2,3]=2 then
(ldf3(4),ys:caseof22(dum)),
if ys#f then return(ys) else (hgstdd(dum),
ldf3(5),gsvhg(dum),return(ys)))$
pfdivide(y0,i,lamu):=block([], wy:ratsimp(y0*(x-la[i,0])^lamu),
la[i,1]:ratsimp(la[i,1]-lamu), la[i,2]:ratsimp(la[i,2]-lamu),
la[3,1]:ratsimp(la[3,1]+lamu), la[3,2]:ratsimp(la[3,2]+lamu),
v:-la[3,1], return(wy))$
listofpars(exp):=block([pl],pl:listofvars(exp), pl:delete(x,pl),
pl:delete(y,pl), lplist:length(pl), return(pl))$
contratn(i):=block([], dexp:dla[i], j:1,
wexp:ratsimp(dexp), if wexp=-1/2 then return(f),
dnop:ratsimp(dexp), if integerp(dnop) then return(dnop),
loop, dnop:ratsimp(dexp-1/j), if psdmint(dnop) then return(1/j),
j:j+1, if j<6 then go(loop),
dnop:ratsimp(dexp-2/5), if psdmint(dnop) then return(2/5),
return(f))$
psdmint(dnop):=block([],wdno:ratsimp(dnop),
if integerp(wdno) then (if wdno<=0 then return(t) else return(f)),
if lplist=0 then return(f),
if lplist=1 then
(if denom(wdno)#1 then return(f),
ans:read("Is",wdno,"a minus integer?",str4),
if ans=y then return(t) else return(f)))$
sortratn(dum):=block([], nrat:0,
for i:1 thru 3 do (if cdla[i]#f then
(nrat:nrat+1, la[i,3]:denom(cdla[i]), if la[i,3]=1 then
(if cdla[i]=0 then la[i,3]:0 else (minla[i]:min(abs(la[i,1]),abs(la[i,2])),
la[i,3]:la[i,3]+minla[i])))
else (if la[i,1]*la[i,2]=0 then la[i,3]:8 else la[i,3]:9)),
clsort(dum))$
exccolumns(dum):=block([],
if la[1,4]#1 then (if la[2,4]=1 then exc2c(1,2) else exc2c(1,3)),
if la[2,4]=3 then exc2c(2,3),
for i:1 thru 3 do dla[i]:ratsimp(la[i,1]-la[i,2]))$
exc2c(l,m):=block([],for i:0 thru 4 do exchla(i,l,m))$
exchla(i,l,m):=block([],ws:la[l,i],la[l,i]:la[m,i],la[m,i]:ws)$
wlength(exp):=block([], p1:first(plist),
if not freeof(p1,exp) then (wdeg:hipow(exp,p1),return(wdeg))
else return(0))$
mklntr(x1,x2,x3):=block([a,b,c,d],
if x3=inf then (a:1/(x2-x1),b:-a*x1, c:0,d:1 ) else
if x2=inf then (a:1, b:-a*x1, c:1,d:-x3) else
if x1=inf then (a:0, b:x2-x3 ,c:1,d:-x3),
vtr:t=ratsimp((a*x+b)/(c*x+d)) )$
checkcontratn(i):=block([wla],
cdla[i]:contratn(i), if cdla[i]#f then return(t),
exc1c(i), cdla[i]:contratn(i))$
hgstdd(dum):=block([],
if la[1,0]=0 and la[2,0]=1 and la[3,0]=inf then return(f),
mklntr(la[1,0],la[2,0],la[3,0]),
print(str2,vtr), la[1,0]:0,la[2,0]:1,la[3,0]:inf,
print(str1),pfuncthg(n,dum))$
make01p(i):=block([],
if la[i,1]=la[i,2] and la[i,2]#0 then y0:pfdivide(y0,i,la[i,2]) else
(l1:wlength(la[i,1]),l2:wlength(la[i,2]),
if l1>l2 then excrow(dum),
if la[i,1]#0 then y0:pfdivide(y0,i,la[i,2]),
if la[i,2]=0 then exc1c(1)))$
clsort(dum):=block([], for i:1 thru 3 do la[i,4]:3,
for i:1 thru 3 do (i1:nmod(i+1,3),i2:nmod(i+2,3),
if la[i ,3]<=la[i1,3] and la[i,3]<=la[i2,3] then (la[i,4]:1,
if la[i1,3]<=la[i2,3] then la[i1,4]:2 else la[i2,4]:2)))$
nmod(n,k):=block([],if n>k then return(n-k) else return(n))$
LODESave([phghp,fasl],haspmts,pfdivide,listofpars,contratn,
checkcontratn,mklntr,psdmint,sortratn,exccolumns,
exc2c,exchla,wlength,hgstdd,make01p,clsort,nmod);
gsvlgdre(v):=block([],remvalue(L),
print("The solution is representable by the solution of Legendre's eq:
(x^2-1)*y''+2*x*y'-v*(v+1)*y=0"),
if la[1,2]=0 and la[2,2]=0 then svlgdre(v) else
(if la[1,2]=la[2,2] then
(if integerp(la[1,2]) then
(if la[1,2]>0 then (y0:y0*expand(xa*xb),
la[3,1]:la[3,1]+la[1,2], la[3,2]:la[3,2]+la[1,2], v:-la[3,2],
yw:y[L](v,x), ys:y0^la[1,2]*'diff(yw,x,la[1,2]),
print("y=",ys), return(ys)))
else
(ans:read("Is",la[1,2],str5),
if ans=p then (w0:expand(xa*xb),y0:y0*w0^la[1,2] ,
la[3,1]:la[3,1]+la[1,2], la[3,2]:la[3,2]+la[1,2],v:-la[3,2]) else
if ans=m then (la[1,2]:-la[1,2], la[3,1]:la[3,1]-la[1,2],
la[3,2]:la[3,2]-la[1,2], v:-la[3,2]),
remvalue(L), yw:y[L](v,x), ys:y0*'diff(yw,x,la[1,2]),
print("y=",ys,",where y[L](v,x) is the solution of Legendre's eq."),
return(ys)) ) ))$
svlgdre(v):=block([],
if spt[1]*spt[2]=-1 and (spt[1]=1 or spt[1]=-1) then
(ys:y0*y[L](v,x), print("y=",ys), return(ys))
else (lfrtr:(-2*x+spt[1]+spt[2])/(spt[1]-spt[2]),lfrtr:ratsimp(lfrtr),
vtr:t=lfrtr, ys:y0*y[L](v,t),
print("y=",ys,"where t=",lfrtr),return(ys)))$
LODESave([psleg,fasl],svlgdre,gsvlgdre);
caseof22(dum):=block([], ans:no,
mcdla:ratsimp(-dal[1]+1/2),
if not integerp(mcdla) then
ans:read("Is",mcdla,"a positive integer?",str4),
if (integerp(mcdla) and mcdla>=0) or ans=y then (
w1:sqrt(xa), w2:sqrt(xb),
wy1:(w1+w2)^dla[3], wy2:(w1-w2)^dla[3],
ys:y0*'diff(k1*wy1+k2*wy2,x,mcdla),print("y=",ys),return(ys))
else return(f))$
LODESave([phyp25,fasl],caseof22);
gsvhg(dum):=block([E,K], ys:f, remvalue(E,K),
if la[1,0]=0 and la[2,0]=1 and la[1,1]=0 and la[2,1]=0 then
(print("The solution is representable by Hypergeometric function."),
ha:la[3,1],hb:la[3,2],hg:-la[1,2]+1, remvalue(G), ys:y0*y[G](ha,hb,hg,x),
if ha=-1/2 and hb=1/2 and hg=1 then
(y1:E(sqrt(x)),y2:E(sqrt(1-x))-K(sqrt(1-x)),ys:k1*y1+k2*y2,print("y=",ys),
print("where E and K are elliptic functions of 1st and 2nd kind."))
else print("y=",ys), return(ys)))$
LODESave([psolh,fasl],gsvhg);
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