Particular solutions and integrating factors of some ODEs of any order Yu.N. KosovtsovLviv Radio Engineering Research Institute, UkraineEmail: kosovtsov@escort.lviv.netCopyright, 2005
<Text-field layout="Heading 1" style="Heading 1">Abstract</Text-field>Very often given ODE defies solution. In such situations obtaining exact particular solutions is very desirable. There are only a few sufficiently general methods for obtaining the particular solutions. The most popular is simple guessing. But on this way we can look over only simplest hypothesis.
In [ http://www.maplesoft.com/applications/app_center_view.aspx?AID=1309 ] we propose a new algebraic method for finding integrating factors of some ODEs of any order. We also have demonstrated there that there are fast CAS procedures which can obtain components of an integrating factor. It is well-known that some components of an integrating factor coincide with some components of a first integral. So this method can be applied to obtaining the particular solutions of some ODEs of any order. With help of modified demonstration procedure koif we show that the method can be fruitful. Our procedure is workable for rational type non-linear ODEs (and surprisingly for many transcendental ODEs even with symbolic parameters) of orders from n=1 to n=5-10. The procedure can handle some simple ODEs of very high order (n>NiMqJCIjNSIiJA==).<Text-field layout="Heading 1" style="Heading 1">1. The method</Text-field>
1 Introduction
Very often given ODE defies solution. In such situations obtaining exact particular solutions is very desirable.
There are only a few sufficiently general methods for obtaining the particular solutions. The most popular is simple guessing. But on this way we can look over only simplest hypothesis.
More productive is the usage of the first integrals. But for many interesting real-world problems we are not able to obtain full structure of a first integral. Nevertheless, if we would be able to find certain components of a first integral, we would to proceed particular solutions hunting.
In [1] we propose a new algebraic method for finding integrating factors of some ODEs of any order. We also have demonstrated in [1] that there are fast CAS procedures which can obtain some (or even all) components of an integrating factor. It is well-known that some components of an integrating factor coincide with some components of a first integral. So this method can be applied to obtaining the particular solutions of some ODEs of any order.With help of modified (with respect to [1]) demonstration procedure koif (see below) we show that the method can be fruitful.
2 The base of the method
For nth order ODE in solved form NiMvLCYtJSVkaWZmRzYkJiUieUc2IyIiIS0lIiRHNiQlInhHJSJuRyIiIi0lImZHNiVGL0YoOyZGKTYjRjEmRik2IywmRjBGMUYxISIiRjtGKw== (1)where NiMvJiUieUc2IyIiIS1GJTYjJSJ4Rw== , NiMvJiUieUc2IyUiakctJSVkaWZmRzYkLUYlNiMlInhHLSUiJEc2JEYtRic=by standard definition [see,e.g.,2] NiMlI211Rw== is an integrating factor if NiMqJiUjbXVHIiIiLCYmJSJ5RzYjJSJuR0YlJSJmRyEiIkYl is a total derivative of some function NiMtJSV6ZXRhRzYlJSJ4RyYlInlHNiMiIiE7JkYoNiMiIiImRig2IywmJSJuR0YuRi4hIiI= , that is NiMvKiYlI211RyIiIiwmLSUlZGlmZkc2JCYlInlHNiMiIiEtJSIkRzYkJSJ4RyUibkdGJiUiZkchIiJGJiwmLUYpNiQlJXpldGFHRjJGJi0lJHN1bUc2JComJkYsNiMsJiUiakdGJkYmRiZGJi1GKTYkRjkmRiw2I0ZBRiYvRkE7Ri4sJkYzRiZGJkY1RiY= . (2)Let NiMvJSNtdUctJSVkaWZmRzYkJSV6ZXRhRyYlInlHNiMsJiUibkciIiJGLiEiIg== (3)then (2) becomesNiMvLCYtJSJERzYjJSV6ZXRhRyIiIiomJSJmR0YpLSUlZGlmZkc2JEYoJiUieUc2IywmJSJuR0YpRikhIiJGKUYpIiIh (4)where NiMvLSUiREc2IyUlemV0YUcsJi0lJWRpZmZHNiRGJyUieEciIiItJSRzdW1HNiQqJiYlInlHNiMsJiUiakdGLUYtRi1GLS1GKjYkRicmRjM2I0Y2Ri0vRjY7IiIhLCYlIm5HRi0iIiMhIiJGLQ== .So we conclude that the integrating factor of ODE (1) is NiMvJSNtdUctJSVkaWZmRzYkJSV6ZXRhRyYlInlHNiMsJiUibkciIiJGLiEiIg== , where the function NiMlJXpldGFH satisfies the linear first-order PDE (4).
Let us suppose that PDE (4) has some solution in the following formNiMvJSV6ZXRhRyomLSUocHJvZHVjdEc2JCkmJSJQRzYjJSJqRyYlJmFscGhhR0YsL0YtOyIiIiUjbjFHRjItRic2JCkmJSJRRzYjJSJrRyYlJWJldGFHRjkvRjo7RjIlI24yRyEiIg==
(5)where NiMmJSJQRzYjJSJqRw== and NiMmJSJRRzYjJSJrRw== (for the sake of simplicity) are polynomials, NiMmJSZhbHBoYUc2IyUiakc= and NiMmJSViZXRhRzYjJSJrRw== are constants. So there exists the corresponding integrating factor (3) with the following structure (see [1])NiMvJSNtdUcqJi0lJmRlbm9tRzYjJSJmRyIiIi0lKHByb2R1Y3RHNiQpJiUiUUc2IyUia0csJiYlJWJldGFHRjFGKkYqRiovRjI7RiolI24yRyEiIg== (6)or if exist some NiMyJiUmYWxwaGFHNiMlImlHIiIi ( NiMyIiIhJiUmYWxwaGFHNiMlImlH )NiMvJSNtdUcqJi0lJmRlbm9tRzYjJSJmRyIiIiomLSUocHJvZHVjdEc2JComMiYlJmFscGhhRzYjJSJpR0YqRiopJiUiUEdGMywmRipGKkYxISIiRiovRjQ7RiolI24xR0YqLUYtNiQpJiUiUUc2IyUia0csJiYlJWJldGFHRkJGKkYqRiovRkM7RiolI24yR0YqRjk=
(7)As NiMlJXpldGFH is the so-called first integral for ODE (1) then standard solving strategy consist in reducing the given ODE order by equating the first integral to arbitrary constant, say _C. Reduced ODE has the following formNiMvJSV6ZXRhRyUjX0NH (8)In such an approach we must know the full structure of a first integral or (almost equivalently) the full structure of an integrating factor. Alas, for many interesting real-world problems we are not able to obtain neither.
Nevertheless, as it was shown in [1] we can find closed system of equations for components of an integrating factor (or of a first integral) NiMmJSJQRzYjJSJqRw== and NiMmJSJRRzYjJSJrRw== and further obtain explicit expressions for some NiMmJSJQRzYjJSJqRw== and NiMmJSJRRzYjJSJrRw== purely by an algebraical method. It allows us to find reduced ODEs in the following formNiMvJiUiUEc2IyUiakciIiE= or NiMvJiUiUUc2IyUia0ciIiE= , (9)what is equivalent to (8) with NiMvJSNfQ0ciIiE= or NiMvKiYiIiJGJSUjX0NHISIiIiIh , i.e., just to equations for particular solution of given ODE. If this reduced ODEs can be solved, then we obtain (a set of) particular solutions of given ODE.
We have to note here that closed system of equations for components of a first integral gives us only necessary conditions for candidates to be such components. So we have to verify each solution of reduced ODE (or reduced ODE itself) to be correct.Unfortunately not all reduced ODEs can be solved. But as a rule they are simpler appreciably than original ODE.3 Conclusion
The aim here has been to present the simple and relatively fast method for finding particular solutions of some rational high order non-linear ODEs. The experimentations with the procedure implemented in Maple confirmed consistence and efficiency of proposed method for sufficiently high order ODEs. We can conclude that our procedure is workable for rational ODEs (and surprisingly for many transcendental ODEs even with symbolic parameters) of orders from n=1 to n=5-10. The procedure can handle some simple ODEs of very high order (n>NiMqJCIjNSIiJA==). This method allows Maple to extend their solving abilities to solve high order ODEs missed by their own in-house solvers.
References
[1] Yu.N. Kosovtsov, 2 procedures for finding integrating factors of some ODEs of any order. Maple Apps.Centre, http://www.maplesoft.com/applications/app_center_view.aspx?AID=1642 , (2004).
[2] P.J. Olver, Applications of Lie Groups to Differential Equations, Springer-Verlag, New York, (1993).
<Text-field layout="Heading 1" style="Heading 1">2. Procedures</Text-field><Text-field layout="Heading 2" style="Heading 2">Procedures Definitions</Text-field>koif -looks for an integrating factor for (rational type) non-linear ODEs of any orderpartially_red_ode - looks for reduced ODEs for particular solutions to (rational type) ODEs of any order
Calling Sequenceskoif(ode,y(x),con);partially_red_ode(ode,y(x)); Note: The procedure partially_red_ode uses the procedure koif.Parametersode - ordinary differential equation (any order). y(x) - ODE's dependent variable. The first argument of y(x) must be ODE`s independent variable.con - (optional) if there stands str the procedure is ended after calculation of hypothesis of integrating factor structure only.Output koif - computed integrating factor(s) of given ODE. The output may contain a set of integrating factors or integrating factor with arbitrary parameters NiMmJSNfWEc2IyUiaUc=with option str - hypothesis of integrating factor structure ;partially_red_ode - computed set of reduced ODEs for particular solutions to given ODE. restart;with(ODETools):<Text-field layout="Heading 2" style="Heading 2">koif</Text-field>koif :=proc(ode,y,con)
local f1,nu,num,d,den,Pro_Cand_Sel,Cand_Sel,Cand_Sel_fin,A,B,B1,B2,B3,B4,BOB,F,F1,F2,N,d1,d2,nu1,nu2,a_1,a_2,aun,ar1,ar10,ard4,a1,d3,d10,din,did,din1,did1,f,g,g1,j,j2,j0,i,i2,ik,k,kk,kmm,km,r,s,ss,t,u,n,_x,_Y_,hyp1,mu,mu1,ode1,unn,sols2,Largs,Sargs;
option `Copyright (c) 2004-2005 by Yuri N. Kosovtsov. All rights reserved.`;
Pro_Cand_Sel :=proc(Q,P,ar1,Sargs,h)
local a_a,i,ii,j,CAR,KK,KK1,KK2,RR;
KK :={};
KK2 :={};
try
RR :=[eliminate({Q, P},ar1)];
if nops(RR)>0 then
for j from 1 to nops(RR) do
CAR :=numer(factor(a_a*lhs(ODETools[remove_RootOf]((lhs(op(1,op(1,op(j,RR))))-rhs(op(1,op(1,op(j,RR)))))))));
if nops(CAR)>1 then
KK1 :=CAR;
if type(KK1,`*`)=true then
for i from 1 to nops(KK1) do
if has(op(i,KK1),RootOf)=false and select(type,indets(op(i,KK1)),name) intersect Sargs <> {} then
KK2 :=KK2 union {op(i,KK1)}
fi;
od;
for ii from 1 to nops(KK2) do
if type(op(ii,KK2),`^`)=true then
KK :=KK union {op(1,op(ii,KK2))} else KK :=KK union {op(ii,KK2)};
fi;
od;
else KK :=KK union {KK2}
fi;
fi;
if h<>1 then
KK2 :={};
if op(2,op(j,RR))<>{} then
KK1 :=factor(a_a*op(1,op(2,op(j,RR))));
if type(KK1,`*`)=true then
for i from 1 to nops(KK1) do
if select(type,indets(op(i,KK1)),name) intersect Sargs <> {} then
KK2 :=KK2 union {op(i,KK1)}
fi;
od;
for ii from 1 to nops(KK2) do
if type(op(ii,KK2),`^`)=true then
KK :=KK union {op(1,op(ii,KK2))} else KK :=KK union {op(ii,KK2)};
fi;
od;
else KK :=KK union {KK2}
fi;
fi;
fi;
od;
fi;
catch:
end try;
RETURN(KK)
end proc:
Cand_Sel :=proc(Q,nu,d,Largs,n)
local a_a,a,ar11,i,k,j,sols1,R1,DD,EQ1,YY,W,Can,Cand,RD_Cand;
Cand :={};
try
if nops(Q)<>0 then
YY :={seq(Y[jj], jj=1..nops(Q))};
W :=add(Y[ik]*op(ik,Q), ik=1..nops(Q));
RD_Cand :=diff(W,op(nops(Largs),Largs));
DD :=diff(W,op(1,Largs))+add(op(gg+2,Largs)*diff(W,op(gg+1,Largs)),gg=1..n-1);
a :=select(type,indets(W),name) intersect convert(Largs,set);
if a<>{}then
for i from 1 to nops(a) do
ar11 :=op(i,a);
R1 :=ar11=RootOf(W,ar11);
EQ1 :=numer(factor(subs(R1,nu*RD_Cand+d*DD)));
sols1 :={solve({EQ1},YY)};
for k from 1 to nops(sols1) do
if has(op(k,sols1),Largs)=false then
Can :=factor(a_a*subs(seq(Y[jjj]=1,jjj=1..nops(Q)),subs(op(k,sols1),W)));
for j from 1 to nops(Can) do
if (select(type,indets(op(j,Can)),name) intersect convert(Largs,set))<>{} then
Cand := Cand union {op(j,Can)};
fi;
od;
fi;
od;
od;
fi;
fi;
catch:
end try;
RETURN(Cand)
end proc:
Cand_Sel_fin := proc(F, nu,d,n)
local a,aa,j,v,B2,Br,EQ,dd,jj,R1;
B2 :={};
if nops(F)<>0 then
for j to nops(F) do
try
Br :=op(j,F);
a :=diff(Br,op(n+1,Largs));
dd := diff(Br,op(1,Largs))+add(op(g+2,Largs)*diff(Br,op(g+1,Largs)),g=1..n-1);
aa :=select(type,indets(Br),name) intersect convert(Largs,set);
for v from 1 to nops(aa) do
R1 :=op(v,aa)=RootOf(Br,op(v,aa));
EQ :=simplify(numer(factor(subs(R1,nu*a+d*dd))));
if EQ=0 then B2 :=B2 union {Br} fi;
od;
catch:
end try;
od;
fi;
RETURN(B2);
end proc:
mu:={};
_x :=op(1,y);
n :=PDEtools[difforder](select(has,indets(ode),y));
_Y_ :=op(0,y);
Largs :=[_x,seq(_Y_[m],m=0..n-1)];
Sargs :=convert(Largs,set);
F :=subs({seq(diff(y,`$`(_x,i0))=_Y_[i0],i0=1..n),y=_Y_[0]},ode);
f :=[solve(F,_Y_[n])];
for j0 from 1 to nops(f) do
f1 :=op(j0,f);
nu :=numer(f1);
d :=denom(f1);
N :=(length(nu)+length(d))/2;
B :={};
if type(nu,`*`)=true then nu1 :={op(nu)} else nu1 :={nu}; fi;
nu2 :=nu1;
for i to nops(nu1) do
if select(type,indets(op(i,nu1)),name)={op(n+1,Largs)} then
nu2 :=nu2 minus {op(i,nu1)};
if type(op(i,nu1),`^`)=true then B :=B union {op(1,op(i,nu1))};
else B :=B union {op(i,nu1)};
fi;
fi;
od;
num :=convert(nu2,`*`);
if type(d,`*`)=true then d1 :={op(d)}; else d1 :={d}; fi;
d2 :=d1;
for j to nops(d1) do
if type(op(j,d1),constant)=true then d2 :=d2 minus {op(j,d1)}; fi;
if select(type,indets(op(j,d1)),name) intersect {op(n+1,Largs)}={}
and type(op(j,d1),constant)=false then
if type(op(j,d1),`^`)=true then B :=B union {op(1,op(j,d1))};
else B :=B union {op(j,d1)};
fi;
fi;
od;
den :=convert(d2,`*`);
#1
B1 :={};
B2 :={};
B3 :={};
for k from 1 to nops(d2) do
ar10 :=select(type,indets(op(k,d2)),name);
if ar10<>{} then
for km from 1 to nops(ar10) do
ar1 :=op(km,ar10);
a_1 :=Pro_Cand_Sel(op(k,d2),num,ar1,Sargs,0);
a1 :=sort(convert(a_1,list),length);
if nops(a1)<3 then r :=nops(a1) else r :=nops(a1)-1 fi;
for kk from 1 to r do
if select(type,indets(op(kk,a1)),name) intersect Sargs <>{}then
g1 :={};
if length(op(kk,a1))<N then g1 :=g1 union {op(kk,a1)} fi;
B3 :=B3 union g1;
fi;
od;
od;
fi;
od;
B1 :=B1 union Cand_Sel_fin(B3, nu,d,n);
B :=B union B1;
B1 :=B1 union B3;
nops(B); userinfo(2,{koif}, `1st round is performed, ` ||%|| ` candidates are obtained`);
userinfo(3,{koif}, B);
#2
B4 :={};
for s from 1 to nops(B1) do
B2 :={};
B3 :={};
ar10 :=((select(type,indets(op(s,B1)),name) intersect select(type,indets(num),name) )intersect select(type,indets(d),name)) intersect Sargs;
if ar10 <>{} then
for kmm from 1 to nops(ar10) do
ar1 :=op(kmm,ar10);
a_1 :=Pro_Cand_Sel(op(s,B1),num,ar1,Sargs,0);
a_2 :=Pro_Cand_Sel(op(s,B1),d,ar1,Sargs,0);
aun :=Pro_Cand_Sel(op(s,B1),num,ar1,Sargs,1) union Pro_Cand_Sel(op(s,B1),d,ar1,Sargs,1);
a1 :=a_1 intersect a_2;
for ss from 1 to nops(a1) do
if select(type,indets(op(ss,a1)),name) intersect {op(n+1,Largs)} <>{} then
g :={op(ss,a1)};
g1 :={};
for i2 from 1 to nops(g) do
if length(op(i2,g))<N/2 then g1 :=g1 union {op(i2,g)} fi;
od;
B3 :=B3 union g1;
fi;
od;
for j from 1 to nops(aun) do
if select(type,indets(op(j,aun)),name) intersect {op(n+1,Largs)} <>{} then
if length(op(j,aun))<N/2 then g1 :=g1 union {op(j,aun)} fi;
B2 :=B2 union g1;
fi;
od;
od;
fi;
B2 :=B2 union Cand_Sel(B3,nu,d,Largs,n);
B4 :=B4 union Cand_Sel_fin(B2 union B3, nu,d,n);
od;
B :=B union B4;
BOB :=B;
for t from 1 to nops(B) do
if {-op(t,B)} intersect (BOB minus {op(t,B)})<>{} then BOB :=BOB minus {op(t,B)} fi;
od;
B :=BOB;
nops(B); userinfo(2,{koif}, `2nd round is performed, ` ||%|| ` candidates are obtained`);
userinfo(3,{koif}, B);
#3
if B={} then
F1 :={};
F2 :={};
ard4 := Sargs minus {op(n+1,Largs)};
for j2 to nops(ard4) do
din :=factor(a_a*simplify(diff(num,op(j2,ard4))));
did :=factor(a_a*simplify(diff(den,op(j2,ard4))));
if type (din,`*`)=true then din1 :={op(din)}; else din1 :={din}; fi;
if type (did,`*`)=true then did1 :={op(did)}; else did1 :={did}; fi;
for j from 1 to nops(din1) do F1 :=F1 union Cand_Sel_fin({op(j,din1)}, nu,d,n) od;
for j from 1 to nops(did1) do F2 :=F2 union Cand_Sel_fin({op(j,did1)}, nu,d,n) od;
od;
B :=B union F1;
B :=B union F2;
nops(B); userinfo(2,{koif}, `3rd round is performed, ` ||%|| ` candidates are obtained`);
fi;
A :=product(op(l,B)^_X[l],l=1..nops(B));
d10 :=d1;
for u from 1 to nops(d1) do
if type(op(u,d1),`^`)=true then
d2 :={op(1,op(u,d1))} else
d2 :={op(u,d1)}
fi;
if d2 intersect B <>{} then d10:=d10 minus {op(u,d1)} fi;
if select(type,indets(op(u,d1)),name) intersect Sargs ={} then
d10 :=d10 minus {op(u,d1)}
fi;
od;
if d10={} then mu1 :=1/A else d3:=convert(d10,`*`); mu1 :=d3/A; fi;
userinfo(2,{koif}, `Hypothesis of mu structure is found`,mu1);
hyp1 :=subs(seq(_Y_[i3]=diff(y,`$`(_x,i3)),i3=1..(n-1)),_Y_[0]=y,mu1);
ode1 :=diff(y,`$`(_x,n))-subs(seq(_Y_[i3]=diff(y,`$`(_x,i3)),i3=1..(n-1)),_Y_[0]=y,f1);
if args[nargs]<>str then
unn :={seq(_X[l1],l1=1..nops(B))};
sols2 :={solve({ODETools[gensys](ode1, _mu = hyp1,y)},unn)};
for ik from 1 to nops(sols2) do
if has(op(ik,sols2),Sargs union {op(0,y),seq(_Y_||i,i=1..n-1)})=false then
mu:=mu union {subs(op(ik,sols2),hyp1)};
fi;
od;
else
mu:=mu union {int_factor_form=hyp1};
fi;
od;
RETURN(mu)
end proc:
<Text-field layout="Heading 2" style="Heading 2">partially_red_ode</Text-field>partially_red_ode :=proc(ode,y)
local i,d, _x,n,red_ode,mu_str;
option `Copyright (c) 2005 by Yuri N. Kosovtsov. All rights reserved.`;
_x :=op(1,select(type,indets(y),name));
n :=PDEtools[difforder](select(has,indets(ode),y));
red_ode :={};
mu_str :=koif(ode,y,str);
for i from 1 to nops(mu_str) do
d :=denom(rhs(op(i,mu_str)));
red_ode :=red_ode union subs(seq(_X[j]=1,j=1..nops(d)),{op(d)})
od;
red_ode :=select(has, red_ode, diff(y,`$`(_x,n-1)));
RETURN(red_ode)
end proc:Execute p.2 before runing examples!<Text-field layout="Heading 1" style="Heading 1">3. Examples</Text-field>PDEtools[declare](y(x),prime=x):To control the process in progress you may useinfolevel[koif] :=3;with 2 or 3 in right-hand side.<Text-field layout="Heading 1" style="Heading 1">3rd order ODE </Text-field>It is a typical example for the methodode := (-21*diff(y(x),`$`(x,2))^2-14*(x^2+y(x))*diff(y(x),`$`(x,2))+3*a*x+3*diff(y(x),x))*diff(y(x),`$`(x,3)) - ((-14*x-7*diff(y(x),x)+3)*diff(y(x),`$`(x,2))^2+(3*a+x^2+y(x))*diff(y(x),`$`(x,2))+(y(x)-x^2-x*diff(y(x),x))*a-2*x*diff(y(x),x)-diff(y(x),x)^2)=0;The procedure partially_red_ode obtains the following set of reduced ODEs (in sense that
a solution of a reduced ODE is a particular solution of the original ODE)PRODE :=partially_red_ode(ode,y(x));Note: The outputs of different program starts, as a matter of fact, are different, because Maple uses an ordering convenient for its implementation. So the ordering of the following answers can be rearranged.Two of reduced ODEs can be easily solved by Maple and we obtain the following answersans1 :=[dsolve(op(1,PRODE))];ans2 :=[dsolve(op(2,PRODE))];ans3 :=[dsolve(op(3,PRODE))];We have noted above that closed system of equations for components of a first integral gives us only necessary conditions for candidates to be such components. So we have to verify each solution of reduced ODE to be correct.odetest(op(1,ans1),ode);odetest(op(1,ans2),ode);odetest(op(1,ans3),ode);Maple does not find solutions of the third reduced ODE, but it is easy to show that the solutions of this equation are particular solutions of the original ODE too (see, e.g., the following line)PDEtools[casesplit]({ode,op(1,PRODE)});In principle, for this ODE you can try to obtain a set of integrating factors mu :=koif(ode,y(x));ODETools[mutest](op(1,mu),ode);and further find its general solution by standard integrating factor method, but this way requires much more time and memory.<Text-field layout="Heading 1" style="Heading 1">2nd order ODE with trigonometric functions </Text-field>ode := diff(y(x),`$`(x,2)) = -(cos(y(x)+diff(y(x),x))*(x^2+2*x*diff(y(x),x)+beta*diff(y(x),x)^2)+(x+diff(y(x),x))*x*sin(y(x)+diff(y(x),x))*diff(y(x),x)*(beta*diff(y(x),x)+x))/(cos(y(x)+diff(y(x),x))*x*(beta-1)+sin(y(x)+diff(y(x),x))*(x+diff(y(x),x))*(beta*diff(y(x),x)+x))/x;Here we are looking for reduced ODEs for particular solutions to odePRODE :=partially_red_ode(ode,y(x));Maple solves reduced ODEsans1 :=[dsolve(op(1,PRODE))];
ans2 :=[dsolve(op(2,PRODE))];
ans3 :=[dsolve(op(3,PRODE))];
and we have to verify obtained particular solutionsodetest(op(1,ans1),ode,y(x));odetest(op(1,ans2),ode,y(x));odetest(op(1,ans3),ode,y(x));<Text-field layout="Heading 1" style="Heading 1">5th order ODE </Text-field>ode := diff(y(x),`$`(x,5)) = -(1+diff(y(x),x)+diff(y(x),`$`(x,2))+diff(y(x),`$`(x,3))+diff(y(x),`$`(x,4)))*(diff(y(x),x)^2+(2*x+2*diff(y(x),`$`(x,2))+14*diff(y(x),`$`(x,4))+2*y(x)+2*diff(y(x),`$`(x,3)))*diff(y(x),x)+diff(y(x),`$`(x,2))^2+(2*x+14*diff(y(x),`$`(x,4))+2*y(x)+2*diff(y(x),`$`(x,3)))*diff(y(x),`$`(x,2))+diff(y(x),`$`(x,3))^2+(2*x+14*diff(y(x),`$`(x,4))+2*y(x))*diff(y(x),`$`(x,3))+49*diff(y(x),`$`(x,4))^2+(14*x+14*y(x))*diff(y(x),`$`(x,4))+1+x^2+2*x*y(x)+alpha+y(x)^2)/(3*diff(y(x),x)^2+(6*x+6*diff(y(x),`$`(x,2))+42*diff(y(x),`$`(x,4))+6*y(x)+6*diff(y(x),`$`(x,3)))*diff(y(x),x)+3*diff(y(x),`$`(x,2))^2+(6*x+42*diff(y(x),`$`(x,4))+6*y(x)+6*diff(y(x),`$`(x,3)))*diff(y(x),`$`(x,2))+3*diff(y(x),`$`(x,3))^2+(6*x+42*diff(y(x),`$`(x,4))+6*y(x))*diff(y(x),`$`(x,3))+147*diff(y(x),`$`(x,4))^2+(42*x+42*y(x))*diff(y(x),`$`(x,4))+3+3*x^2+6*x*y(x)+7*alpha+3*y(x)^2);Here we are looking for reduced ODEs for particular solutions to odePRODE :=partially_red_ode(ode,y(x));Maple solves reduced ODEans :=[dsolve(op(1,PRODE))];
and we have to verify obtained particular solutions, but this may require much time and memory.odetest(op(1,ans),ode,y(x));<Text-field layout="Heading 1" style="Heading 1">1150th order ODE</Text-field>OFF;ode := diff(y(x),`$`(x,1150)) = (((14*alpha-3)*diff(y(x),`$`(x,1149))+2*alpha*x-alpha*y(x)-3*x^2)*diff(y(x),x)+(28*x-3)*diff(y(x),`$`(x,1149))+x^2+4*x*y(x)-3*alpha*y(x))/(-2*x-2*y(x)+7*diff(y(x),`$`(x,1149))+21*x^2+21*alpha*y(x));PRODE :=partially_red_ode(ode,y(x));Maple can solve reduced ODEs, but it requires plenty of time and space to displayans :=dsolve(op(1,PRODE));odetest(ans,ode);ON;Legal Notice: The copyright for this application is owned by the author(s). Neither Maplesoft nor the author are responsible for any errors contained within and are not liable for any damages resulting from the use of this material. This application is intended for non-commercial, non-profit use only. Contact the author for permission if you wish to use this application in for-profit activities.