\^/ Maple V Release 4 (WM  Internal Use Only)
._\ /_. Copyright (c) 19811996 by Waterloo Maple Inc. All rights
\ MAPLE / reserved. Maple and Maple V are registered trademarks of
<____ ____> Waterloo Maple Inc.
 Type ? for help.
>
> infolevel[dsolve]:=10:
# firstord1
> dsolve((x^4x^3)*diff(u(x),x) + 2*x^4*u(x) = x^3/3 + C,u(x));
dsolve/diffeq/dsol1: > first order, first degree methods :
dsolve/diffeq/dsol1: trying linear bernoulli
dsolve/diffeq/linearsol: solving 1st order linear d.e.
bytes used=2000832, alloc=1703624, time=0.97
dsolve/diffeq/dsol1: linear bernoulli successful
3 2 2
(1/6 exp(2 x) x  1/4 exp(2 x) x + 1/2 C exp(2 x) + _C1 x ) exp(2 x)
u(x) = 
2 2
(1  2 x + x ) x
# firstord2
> dsolve(1/2*diff(u(x),x)+u(x)=sin(x),u(x));
dsolve/diffeq/dsol1: > first order, first degree methods :
dsolve/diffeq/dsol1: trying linear bernoulli
dsolve/diffeq/linearsol: solving 1st order linear d.e.
dsolve/diffeq/dsol1: linear bernoulli successful
u(x) = 2/5 cos(x) + 4/5 sin(x) + exp(2 x) _C1
# firstord3
> dsolve(diff(y(x),x)=y(x)/(y(x)*log(y(x))+x),y(x));
dsolve/diffeq/dsol1: > first order, first degree methods :
dsolve/diffeq/dsol1: trying linear bernoulli
dsolve/diffeq/dsol1: trying separable
dsolve/diffeq/dsol1: trying exact
dsolve/diffeq/inexsol: finding solution to inexact d.e.
dsolve/diffeq/exactsol: finding solution to exact d.e.
dsolve/diffeq/dsol1: exact successful
x 2
  1/2 ln(y(x)) = _C1
y(x)
# firstord4
> dsolve(2*y(x)*diff(y(x),x)^22*x*diff(y(x),x)y(x)=0,y(x));
dsolve/diffeq/clairchk: determining if d.e. is Clairaut
dsolve/diffeq/foxsol: solving high degree d.e. for x
dsolve/diffeq/foysol: solving high degree d.e. for y
dsolve/diffeq/dsol1: > first order, first degree methods :
dsolve/diffeq/dsol1: trying linear bernoulli
dsolve/diffeq/dsol1: trying separable
dsolve/diffeq/dsol1: trying exact
dsolve/diffeq/dsol1: trying general homogeneous
dsolve/diffeq/genhomsol: finding homogeneous solution
bytes used=4001680, alloc=2883056, time=2.52
bytes used=6002152, alloc=3407248, time=4.31
dsolve/diffeq/dsol1: general homogeneous successful
dsolve/diffeq/dsol1: > first order, first degree methods :
dsolve/diffeq/dsol1: trying linear bernoulli
dsolve/diffeq/dsol1: trying separable
dsolve/diffeq/dsol1: trying exact
dsolve/diffeq/dsol1: trying general homogeneous
dsolve/diffeq/genhomsol: finding homogeneous solution
bytes used=8003216, alloc=3800392, time=6.30
dsolve/diffeq/dsol1: general homogeneous successful
/ 2
 x
x = _C1 x exp( 1/3 x arctanh()
 2 1/2 2 2 1/2
\ (x ) (x + 2 y(x) )
1/2
x (x + 6 y(x))
+ arctanh(1/2 )
2 1/2 2 2 1/2
(x ) (x + 2 y(x) )
1/2 \
x (x  6 y(x))  / 2 1/2 /
+ arctanh(1/2 ) / (x ) ) / (
2 1/2 2 2 1/2  / /
(x ) (x + 2 y(x) ) /
/
2 2 1/3 1/3 
(2 y(x)  3 x ) y(x) ), x = _C1 x exp(1/3 x 

\
2
x
arctanh()
2 1/2 2 2 1/2
(x ) (x + 2 y(x) )
1/2
x (x + 6 y(x))
+ arctanh(1/2 )
2 1/2 2 2 1/2
(x ) (x + 2 y(x) )
1/2 \
x (x  6 y(x))  / 2 1/2 /
+ arctanh(1/2 ) / (x ) ) / (
2 1/2 2 2 1/2  / /
(x ) (x + 2 y(x) ) /
2 2 1/3 1/3
(2 y(x)  3 x ) y(x) )
# bernoulli
> dsolve(diff(y(x),x)+y(x)=y(x)^3*sin(x),y(x));
dsolve/diffeq/dsol1: > first order, first degree methods :
dsolve/diffeq/dsol1: trying linear bernoulli
dsolve/diffeq/bernsol: trying Bernoulli solution
dsolve/diffeq/linearsol: solving 1st order linear d.e.
dsolve/diffeq/dsol1: linear bernoulli successful
1
 = 2/5 cos(x) + 4/5 sin(x) + exp(2 x) _C1
2
y(x)
# bernoulli2
# changed wrt V.3: works only for integer exponent
> dsolve(diff(y(x),x)+P(x)*y(x)=Q(x)*y(x)^17,y(x));
dsolve/diffeq/dsol1: > first order, first degree methods :
dsolve/diffeq/dsol1: trying linear bernoulli
dsolve/diffeq/bernsol: trying Bernoulli solution
dsolve/diffeq/linearsol: solving 1st order linear d.e.
dsolve/diffeq/dsol1: linear bernoulli successful
/ / / \ /
1     
 = 16  exp(16  P(x) dx) Q(x) dx + _C1 exp(16  P(x) dx)
16     
y(x) \ / / / /
> dsolve(diff(y(x),x)+P(x)*y(x)=Q(x)*y(x)^(17),y(x));
dsolve/diffeq/dsol1: > first order, first degree methods :
dsolve/diffeq/dsol1: trying linear bernoulli
dsolve/diffeq/bernsol: trying Bernoulli solution
dsolve/diffeq/linearsol: solving 1st order linear d.e.
dsolve/diffeq/dsol1: linear bernoulli successful
/ / / \ /
18     
y(x) = 18  exp(18  P(x) dx) Q(x) dx + _C1 exp(18  P(x) dx)
    
\ / / / /
# was working for a rational n in V.3
> dsolve(diff(y(x),x)+P(x)*y(x)=Q(x)*y(x)^(2/3),y(x));
dsolve/diffeq/dsol1: > first order, first degree methods :
dsolve/diffeq/dsol1: trying linear bernoulli
dsolve/diffeq/dsol1: trying separable
dsolve/diffeq/dsol1: trying exact
dsolve/diffeq/dsol1: trying general homogeneous
dsolve/diffeq/dsol1: trying Riccati
dsolve: Warning: no solutions found
# but not with a general n
> assume(n>1);
> dsolve(diff(y(x),x)+P(x)*y(x)=Q(x)*y(x)^n,y(x));
dsolve/diffeq/dsol1: > first order, first degree methods :
dsolve/diffeq/dsol1: trying linear bernoulli
dsolve/diffeq/dsol1: trying separable
bytes used=10005776, alloc=3931440, time=8.02
dsolve/diffeq/dsol1: trying exact
dsolve/diffeq/dsol1: trying general homogeneous
dsolve/diffeq/dsol1: trying Riccati
dsolve: Warning: no solutions found
> dsolve(diff(y(x),x)+P(x)*y(x)=Q(x)*y(x)^Pi,y(x));
dsolve/diffeq/dsol1: > first order, first degree methods :
dsolve/diffeq/dsol1: trying linear bernoulli
dsolve/diffeq/dsol1: trying separable
dsolve/diffeq/dsol1: trying exact
dsolve/diffeq/dsol1: trying general homogeneous
dsolve/diffeq/dsol1: trying Riccati
dsolve: Warning: no solutions found
# homogeneous
> dsolve(diff(y(x),x)=(2*x^3*y(x)y(x)^4)/(x^42*x*y(x)^3),y(x));
dsolve/diffeq/dsol1: > first order, first degree methods :
dsolve/diffeq/dsol1: trying linear bernoulli
dsolve/diffeq/dsol1: trying separable
dsolve/diffeq/dsol1: trying exact
dsolve/diffeq/dsol1: trying general homogeneous
dsolve/diffeq/genhomsol: finding homogeneous solution
dsolve/diffeq/dsol1: general homogeneous successful
2
_C1 y(x) x
x = 
2 2
(y(x) + x) (y(x)  x y(x) + x )
# adjoint
> dsolve((x^2x)*diff(u(x),x,x)+(2*x^2+4*x3)*diff(u(x),x)+8*x*u(x)=1,u(x));
dsolve/diffeq/polylinearODE: checking Euler equation
dsolve/diffeq/secorder: checking Bessel's equation
dsolve/diffeq/secorder: checking hypergeometric equation
dsolve/diffeq/secorder: checking RiemannPapperitz equation
bytes used=12454208, alloc=4586680, time=9.77
dsolve/diffeq/secorder: trying polynomial solutions to Riccati
dsolve/diffeq/secorder: trying Kovacic's algorithm
bytes used=14454896, alloc=4979824, time=11.69
dsolve/diffeq/secorder: Kovacic's algorithm successful
3 + 2 x _C1 exp(2 x) _C2
u(x) = 1/12  +  + 
2 2 2 2
(1 + x) (1 + x) x (1 + x)
# autonomous
> dsolve(diff(y(x),x,x)diff(y(x),x)=2*y(x)*diff(y(x),x),y(x));
dsolve/diffeq/linsubs: trying linear substitution
dsolve/diffeq/missbody: solving d.e. with missing variable
dsolve/diffeq/dsol1: > first order, first degree methods :
dsolve/diffeq/dsol1: trying linear bernoulli
dsolve/diffeq/linearsol: solving 1st order linear d.e.
dsolve/diffeq/dsol1: linear bernoulli successful
1 + 2 y(x)
arctan()
1/2
(4 _C1  1)
x = 2   _C2
1/2
(4 _C1  1)
# autonomous2
> dsolve(diff(y(x),x,x)/y(x)diff(y(x),x)^2/y(x)^21+1/y(x)^3=0,y(x));
dsolve/diffeq/linsubs: trying linear substitution
dsolve/diffeq/missbody: solving d.e. with missing variable
dsolve/diffeq/dsol1: > first order, first degree methods :
dsolve/diffeq/dsol1: trying linear bernoulli
dsolve/diffeq/dsol1: trying separable
dsolve/diffeq/dsol1: trying exact
dsolve/diffeq/inexsol: finding solution to inexact d.e.
dsolve/diffeq/exactsol: finding solution to exact d.e.
dsolve/diffeq/dsol1: exact successful
bytes used=16455368, alloc=5241920, time=13.91
y(x)
/ 1/2
 y2 6
x =   1/2  dy2  _C2,
 4 4 1/2
/ (3 ln(y2) y2 + y2 + 3 _C1 y2 )
0
y(x)
/ 1/2
 y1 6
x =  1/2  dy1  _C2
 4 4 1/2
/ (3 ln(y1) y1 + y1 + 3 _C1 y1 )
0
# clairaut
# changed wrt V.3: bug fixed in output of singular solutions
> dsolve((x^21)*diff(y(x),x)^22*x*y(x)*diff(y(x),x)+y(x)^21,y(x));
dsolve/diffeq/dsol1: > first order, first degree methods :
dsolve/diffeq/dsol1: trying linear bernoulli
dsolve/diffeq/dsol1: trying separable
dsolve/diffeq/dsol1: trying exact
dsolve/diffeq/dsol1: trying general homogeneous
bytes used=18457488, alloc=5504016, time=15.94
dsolve/diffeq/dsol1: trying Riccati
dsolve/diffeq/linsubs: trying linear substitution
dsolve/diffeq/dsol1: > first order, first degree methods :
dsolve/diffeq/dsol1: trying linear bernoulli
dsolve/diffeq/dsol1: trying separable
dsolve/diffeq/dsol1: trying exact
dsolve/diffeq/dsol1: trying general homogeneous
dsolve/diffeq/dsol1: trying Riccati
dsolve/diffeq/linsubs: trying linear substitution
dsolve/diffeq/clairchk: determining if d.e. is Clairaut
dsolve/diffeq/clairsol: solving Clairaut equation
dsolve/diffeq/clairchk: determining if d.e. is Clairaut
dsolve/diffeq/clairsol: solving Clairaut equation
2 1/2 2 1/2
y(x) = x _C1  (_C1 + 1) , y(x) = x _C1 + (_C1 + 1) ,
1 1
y(x) = , y(x) =  
/ 1 \1/2 / 1 \1/2
   
 2   2 
\ x  1/ \ x  1/
# clairaut2
# changed wrt V.3: no error any more
> dsolve(f(x*diff(y(x),x)y(x))=g(diff(y(x),x)),y(x));
dsolve/diffeq/clairchk: determining if d.e. is Clairaut
dsolve/diffeq/clairsol: solving Clairaut equation
bytes used=20457824, alloc=5766112, time=18.10
D(g)(_T) %1 D(f)(%1) + _T D(g)(_T)
[x = , y(x) = ],
D(f)(%1) D(f)(%1)
y(x) = x _C1  RootOf(f(_Z)  g(_C1))
%1 := RootOf(f(_Z)  g(_T))
# constantcoeff
> dsolve(diff(y(x),x$7)14*diff(y(x),x$6)+80*diff(y(x),x$5)242*diff(y(x),x$4)
> +419*diff(y(x),x$3)416*diff(y(x),x$2)+220*diff(y(x),x)48*y(x)=0,y(x));
dsolve/diffeq/polylinearODE: trying linear constant coefficient
dsolve/diffeq/polylinearODE: linear constant coefficient successful
y(x) = _C1 exp(x) + _C2 exp(2 x) + _C3 exp(4 x) + _C4 exp(3 x) + _C5 exp(x) x
2
+ _C6 exp(x) x + _C7 exp(2 x) x
# delay
# bug still in V.4
> dsolve(diff(y(t),t)+a*y(t1)=0,y(t));
dsolve/diffeq/dsol1: > first order, first degree methods :
dsolve/diffeq/dsol1: trying linear bernoulli
dsolve/diffeq/dsol1: trying separable
dsolve/diffeq/dsol1: trying exact
dsolve/diffeq/exactsol: finding solution to exact d.e.
dsolve/diffeq/dsol1: exact successful
/

a  y(t)(t  1) dt + y(t) = _C1

/
# several
# changed wrt V.3: pointer to pdesolve which works
> dsolve(diff(y(x,a),x)=a*y(x,a),y(x,a));
Error, (in dsolve) Please try pdesolve
> pdesolve(diff(y(x,a),x)=a*y(x,a),y(x,a));
y(x, a) = exp(a x) _F1(a)
# ymissing
> dsolve(diff(y(x),x,x)+2*x*diff(y(x),x)=2*x,y(x));
dsolve/diffeq/polylinearODE: checking Euler equation
dsolve/diffeq/secorder: checking Bessel's equation
dsolve/diffeq/secorder: checking hypergeometric equation
dsolve/diffeq/secorder: checking RiemannPapperitz equation
dsolve/diffeq/secorder: trying polynomial solutions to Riccati
dsolve/diffeq/secorder: polynomial solutions to Riccati successful
y(x) = x + _C1 + _C2 erf(x)
# diff
> dsolve(2*y(x)*diff(y(x),x,x)diff(y(x),x)^2=1/3*(diff(y(x),x)x*diff(y(x),x,x))^2,y(x));
dsolve: Warning: no solutions found
# equidimx
> dsolve(x*diff(y(x),x,x)=2*y(x)*diff(y(x),x),y(x));
dsolve: Warning: no solutions found
# equidimy
> dsolve((1x)*(y(x)*diff(y(x),x,x)diff(y(x),x)^2)+x^2*y(x)^2=0,y(x));
dsolve/diffeq/linsubs: trying linear substitution
dsolve: Warning: no solutions found
# euler
> dsolve(diff(y(x),x$4)4/x^2*diff(y(x),x,x)+8/x^3*diff(y(x),x)8*y(x)/x^4,y(x));
dsolve/diffeq/polylinearODE: checking Euler equation
dsolve/diffeq/polylinearODE: Euler equation successful
_C2 2 4
y(x) = x _C1 +  + _C3 x + _C4 x
x
# exact1st
> dsolve(diff(y(x),x)=(3*x^2y(x)^27)/(exp(y(x))+2*x*y(x)+1),y(x));
dsolve/diffeq/dsol1: > first order, first degree methods :
dsolve/diffeq/dsol1: trying linear bernoulli
dsolve/diffeq/dsol1: trying separable
dsolve/diffeq/dsol1: trying exact
bytes used=22458304, alloc=5766112, time=20.41
dsolve/diffeq/exactsol: finding solution to exact d.e.
dsolve/diffeq/dsol1: exact successful
3 2
x + y(x) x + 7 x + exp(y(x)) + y(x) = _C1
# exact2nd
> dsolve(x*y(x)*diff(y(x),x,x)+x*diff(y(x),x)^2+y(x)*diff(y(x),x)=0,y(x));
dsolve: Warning: no solutions found
# exactnth
> dsolve((1+x+x^2)*diff(y(x),x$3)+(3+6*x)*diff(y(x),x,x)+6*diff(y(x),x)=6*x,y(x));
dsolve/diffeq/polylinearODE: checking Euler equation
dsolve/diffeq/expsols: trying exponential solutions
dsolve/diffeq/expsols: rational solutions successful
4 2
x _C1 _C2 x _C3 x
y(x) = 1/4  +  +  + 
2 2 2 2
1 + x + x 1 + x + x 1 + x + x 1 + x + x
# circle (Nonlinear, 3th order)
# changed wrt V.3: solution is much more complex
> dsolve((diff(y(x),x)^2+1)*diff(y(x),x$3)3*diff(y(x),x)*diff(y(x),x$2)^2,y(x));
dsolve/diffeq/missbody: solving d.e. with missing variable
dsolve/diffeq/missbody: solving d.e. with missing variable
dsolve/diffeq/dsol1: > first order, first degree methods :
dsolve/diffeq/dsol1: trying linear bernoulli
dsolve/diffeq/linearsol: solving 1st order linear d.e.
dsolve/diffeq/dsol1: linear bernoulli successful
bytes used=24458704, alloc=5897160, time=22.71
y(x) = 
1/2 2 1/2 2 1/2 2 1/2 1/2 2
(%1 (_C1 ) + _C2 ln((_C1 ) x + (_C1 ) _C2 + %1 ) _C1 )
/ 1 \1/2 1/2 / 2 1/2
  %1 / (_C1 (_C1 ) ) +
 2 2 2 2 2  /
\ x _C1 + 2 x _C1 _C2 + _C2 _C1  1/
2 1/2 2 1/2 1/2
ln((_C1 ) x + (_C1 ) _C2 + %1 )
/ 1 \1/2 1/2 / 2 1/2
  %1 _C1 _C2 / (_C1 )
 2 2 2 2 2  /
\ x _C1 + 2 x _C1 _C2 + _C2 _C1  1/
+ _C3
2 2 2 2 2
%1 := x _C1  2 x _C1 _C2  _C2 _C1 + 1
# transfBernoulli (Nonlinear, 4th order)
# bug if one replaces 3 by 1 in front of y''
> dsolve( 3*diff(y(x),x$2)*diff(y(x),x$4)5*diff(y(x),x$3)^2 = 0, y(x) );
1/2
(6 x _C1  6 _C2 _C1)
y(x) = 3  + _C3 x + _C4,
2
_C1
1/2
(6 x _C1  6 _C2 _C1)
y(x) = 3  + _C3 x + _C4
2
_C1
# factor
> dsolve(diff(y(x),x)*(diff(y(x),x)+y(x))=x*(x+y(x)),y(x));
dsolve/diffeq/dsol1: > first order, first degree methods :
dsolve/diffeq/dsol1: trying linear bernoulli
dsolve/diffeq/linearsol: solving 1st order linear d.e.
dsolve/diffeq/dsol1: linear bernoulli successful
dsolve/diffeq/dsol1: > first order, first degree methods :
dsolve/diffeq/dsol1: trying linear bernoulli
dsolve/diffeq/linearsol: solving 1st order linear d.e.
dsolve/diffeq/dsol1: linear bernoulli successful
2
y(x) = x + 1 + exp(x) _C1, y(x) = 1/2 x + _C1
# factoring
> dsolve( diff(y(x),x$2)^2  2*diff(y(x),x)*diff(y(x),x$2)
> + 2*y(x)*diff(y(x),x)  y(x)^2 = 0, y(x));
dsolve/diffeq/polylinearODE: trying linear constant coefficient
dsolve/diffeq/polylinearODE: linear constant coefficient successful
dsolve/diffeq/polylinearODE: trying linear constant coefficient
dsolve/diffeq/polylinearODE: linear constant coefficient successful
y(x) = _C1 exp(x) + _C2 exp(x), y(x) = _C1 exp(x) + _C2 exp(x) x
# intcomb
> dsolve({diff(x(t),t)=3*y(t)*z(t),
> diff(y(t),t)=3*x(t)*z(t),
> diff(z(t),t)=x(t)*y(t)},{x(t),y(t),z(t)});
dsolve/diffeq/system/linear: determining if system is linear
dsolve/diffeq/system: cannot solve nonlinear systems
dsolve/diffeq/system/linear: determining if system is linear
dsolve/diffeq/system: cannot solve nonlinear systems
dsolve: Warning: no solutions found
# liouvillian
> dsolve((x^3/2x^2)*diff(y(x),x,x)+(2*x^23*x+1)*diff(y(x),x)
> +(x1)*y(x),y(x));
dsolve/diffeq/polylinearODE: checking Euler equation
dsolve/diffeq/secorder: checking Bessel's equation
dsolve/diffeq/secorder: checking hypergeometric equation
dsolve/diffeq/secorder: checking RiemannPapperitz equation
bytes used=28459624, alloc=6159256, time=27.47
dsolve/diffeq/secorder: trying polynomial solutions to Riccati
dsolve/diffeq/secorder: trying Kovacic's algorithm
bytes used=30459952, alloc=6159256, time=29.71
dsolve/diffeq/expsols: trying exponential solutions
dsolve/diffeq/expsols_padicpartbounded: eqns 1 deg 2
dsolve/diffeq/expsols_padicpartbounded: eqns 1 deg 2
dsolve/diffeq/expsols_padicpartbounded: eqns 1 deg 1
dsolve/diffeq/expsols_padicpartbounded: eqns 1 deg 1
dsolve/diffeq/expsols_solvericcati: max # of trials : 5
bytes used=32460344, alloc=6159256, time=32.24
dsolve/diffeq/expsols: expon. solutions partially successful. Result(s) =
exp(Int((x^2+2*x2)/(x^32*x^2),x))
dsolve/diffeq/expsols_reduceorder: reduction to order 1
bytes used=34461200, alloc=6290304, time=34.75
dsolve/diffeq/expsols_reduceorder: back in order 2
/
 exp(1/x)
_C2 exp( 1/x)   dx
 3/2 1/2
_C1 exp( 1/x) / x (x  2)
y(x) =  + 
1/2 1/2 1/2 1/2
x (x  2) x (x  2)
# intfactors
> dsolve(sqrt(x)*diff(y(x),x,x)+2*x*diff(y(x),x)+3*y(x)=0,y(x));
dsolve/diffeq/linearODE: checking Bessel's equation
/ 2 \
1/2 d  /d \
y(x) = DESol({x  _Y(x) + 2 x  _Y(x) + 3 _Y(x)}, {_Y(x)})
 2  \dx /
\dx /
# interchange
> dsolve(diff(y(x),x)=x/(x^2*y(x)^2+y(x)^5),y(x));
dsolve/diffeq/dsol1: > first order, first degree methods :
dsolve/diffeq/dsol1: trying linear bernoulli
dsolve/diffeq/dsol1: trying separable
dsolve/diffeq/dsol1: trying exact
dsolve/diffeq/inexsol: finding solution to inexact d.e.
dsolve/diffeq/exactsol: finding solution to exact d.e.
dsolve/diffeq/dsol1: exact successful
2 3 3 3 3
 1/2 x exp( 2/3 y(x) )  1/2 y(x) exp( 2/3 y(x) )  3/4 exp( 2/3 y(x) )
= _C1
# lagrange
> dsolve(y(x)=2*x*diff(y(x),x)a*diff(y(x),x)^3,y(x));
dsolve/diffeq/clairchk: determining if d.e. is Clairaut
dsolve/diffeq/foxsol: solving high degree d.e. for x
dsolve/diffeq/foysol: solving high degree d.e. for y
dsolve/diffeq/dsol1: > first order, first degree methods :
dsolve/diffeq/dsol1: trying linear bernoulli
dsolve/diffeq/dsol1: trying separable
dsolve/diffeq/dsol1: trying exact
dsolve/diffeq/inexsol: finding solution to inexact d.e.
dsolve/diffeq/exactsol: finding solution to exact d.e.
dsolve/diffeq/dsol1: exact successful
bytes used=36462072, alloc=6290304, time=37.11
/ 1/3 \2 / 1/3 \4
 %1 x   %1 x 
1/6  + 4  x  3/4 a 1/6  + 4  = _C1,
 a 1/3  a 1/3
\ %1 / \ %1 /
/ 1/3 \2
 %1 x 
 1/12   2  + 1/2 %2 x
 a 1/3 
\ %1 /
/ 1/3 \4
 %1 x 
 3/4 a  1/12   2  + 1/2 %2 = _C1,
 a 1/3 
\ %1 /
/ 1/3 \2
 %1 x 
 1/12   2   1/2 %2 x
 a 1/3 
\ %1 /
/ 1/3 \4
 %1 x 
 3/4 a  1/12   2   1/2 %2 = _C1
 a 1/3 
\ %1 /
/ / 3 2 \1/2\
  96 x  81 y(x) a  2
%1 := 108 y(x) + 12    a
\ \ a / /
/ 1/3 \
1/2  %1 x 
%2 := I 3 1/6   4 
 a 1/3
\ %1 /
# lagrange2
> dsolve(y(x)=2*x*diff(y(x),x)diff(y(x),x)^2,y(x));
dsolve/diffeq/clairchk: determining if d.e. is Clairaut
dsolve/diffeq/foxsol: solving high degree d.e. for x
dsolve/diffeq/foysol: solving high degree d.e. for y
dsolve/diffeq/dsol1: > first order, first degree methods :
dsolve/diffeq/dsol1: trying linear bernoulli
dsolve/diffeq/dsol1: trying separable
dsolve/diffeq/dsol1: trying exact
dsolve/diffeq/inexsol: finding solution to inexact d.e.
dsolve/diffeq/exactsol: finding solution to exact d.e.
dsolve/diffeq/dsol1: exact successful
2 1/2 2 2 1/2 3
(x + (x  y(x)) ) x  2/3 (x + (x  y(x)) ) = _C1,
2 1/2 2 2 1/2 3
(x  (x  y(x)) ) x  2/3 (x  (x  y(x)) ) = _C1
# reduction
> dsolve(diff(y(x),x,x)2*x*diff(y(x),x)+2*y(x)=3,y(x));
dsolve/diffeq/polylinearODE: checking Euler equation
dsolve/diffeq/secorder: checking Bessel's equation
dsolve/diffeq/secorder: checking hypergeometric equation
dsolve/diffeq/secorder: checking RiemannPapperitz equation
dsolve/diffeq/secorder: trying polynomial solutions to Riccati
dsolve/diffeq/secorder: trying Kovacic's algorithm
dsolve/diffeq/secorder: Kovacic's algorithm successful
bytes used=38462432, alloc=6290304, time=39.41
1/2 2
I _C2 (I Pi exp(x ) + x erf(I x) Pi)
y(x) = 3/2 + _C1 x + 
1/2
Pi
# riccati
> dsolve(diff(y(x),x)=exp(x)*y(x)^2y(x)+exp(x),y(x));
dsolve/diffeq/dsol1: > first order, first degree methods :
dsolve/diffeq/dsol1: trying linear bernoulli
dsolve/diffeq/dsol1: trying separable
dsolve/diffeq/dsol1: trying exact
dsolve/diffeq/dsol1: trying general homogeneous
dsolve/diffeq/dsol1: trying Riccati
dsolve/diffeq/polylinearODE: trying linear constant coefficient
dsolve/diffeq/polylinearODE: linear constant coefficient successful
dsolve/diffeq/dsol1: Riccati successful
(_C1 sin(x)  cos(x)) exp(x)
y(x) = 
_C1 cos(x) + sin(x)
# riccati2
> dsolve(diff(y(x),x)=y(x)^2x*y(x)+1,y(x));
dsolve/diffeq/dsol1: > first order, first degree methods :
dsolve/diffeq/dsol1: trying linear bernoulli
dsolve/diffeq/dsol1: trying separable
dsolve/diffeq/dsol1: trying exact
dsolve/diffeq/dsol1: trying general homogeneous
dsolve/diffeq/dsol1: trying Riccati
dsolve/diffeq/dsol1: Riccati successful
2
exp(1/2 x )
y(x) = x + 
1/2 1/2 1/2
_C1 + 1/2 I Pi 2 erf(1/2 I 2 x)
# mriccati
> dsolve({diff(x(t),t)a(t)*(y(t)^2x(t)^2)2*b(t)*x(t)*y(t)2*c*x(t),
> diff(y(t),t)b(t)*(y(t)^2x(t)^2)+2*a(t)*x(t)*y(t)2*c*y(t)},{x(t),y(t)});
dsolve/diffeq/system/linear: determining if system is linear
dsolve/diffeq/system: cannot solve nonlinear systems
dsolve/diffeq/system/linear: determining if system is linear
dsolve/diffeq/system: cannot solve nonlinear systems
dsolve: Warning: no solutions found
# scaleinv
> dsolve(x^2*diff(y(x),x,x)+3*x*diff(y(x),x)+2*y(x)1/y(x)^3/x^4,y(x));
dsolve: Warning: no solutions found
# separable
> dsolve(diff(y(x),x)=(9*x^8+1)/(y(x)^2+1),y(x));
dsolve/diffeq/dsol1: > first order, first degree methods :
dsolve/diffeq/dsol1: trying linear bernoulli
dsolve/diffeq/dsol1: trying separable
dsolve/diffeq/sepsol: solving separable d.e.
dsolve/diffeq/dsol1: separable successful
3 9
y(x) + 1/3 y(x)  x  x = _C1
# solvablex
> dsolve(2*x*diff(y(x),x)+y(x)*diff(y(x),x)^2y(x),y(x));
dsolve/diffeq/clairchk: determining if d.e. is Clairaut
dsolve/diffeq/foxsol: solving high degree d.e. for x
dsolve/diffeq/foysol: solving high degree d.e. for y
dsolve/diffeq/dsol1: > first order, first degree methods :
dsolve/diffeq/dsol1: trying linear bernoulli
dsolve/diffeq/dsol1: trying separable
dsolve/diffeq/dsol1: trying exact
dsolve/diffeq/dsol1: trying general homogeneous
dsolve/diffeq/genhomsol: finding homogeneous solution
bytes used=40463096, alloc=6421352, time=41.57
dsolve/diffeq/dsol1: general homogeneous successful
dsolve/diffeq/dsol1: > first order, first degree methods :
dsolve/diffeq/dsol1: trying linear bernoulli
dsolve/diffeq/dsol1: trying separable
dsolve/diffeq/dsol1: trying exact
dsolve/diffeq/dsol1: trying general homogeneous
dsolve/diffeq/genhomsol: finding homogeneous solution
dsolve/diffeq/dsol1: general homogeneous successful
2
x
x arctanh()
2 1/2 2 2 1/2
(x ) (x + y(x) )
_C1 x exp()
2 1/2
(x )
x = ,
y(x)
2
x
x arctanh()
2 1/2 2 2 1/2
(x ) (x + y(x) )
_C1 x exp( )
2 1/2
(x )
x = 
y(x)
# solvabley
> dsolve(xy(x)*diff(y(x),x)+x*diff(y(x),x)^2,y(x));
dsolve/diffeq/clairchk: determining if d.e. is Clairaut
dsolve/diffeq/foxsol: solving high degree d.e. for x
dsolve/diffeq/foysol: solving high degree d.e. for y
dsolve/diffeq/dsol1: > first order, first degree methods :
dsolve/diffeq/dsol1: trying linear bernoulli
dsolve/diffeq/dsol1: trying separable
dsolve/diffeq/dsol1: trying exact
bytes used=42463448, alloc=6421352, time=43.76
dsolve/diffeq/dsol1: trying general homogeneous
dsolve/diffeq/genhomsol: finding homogeneous solution
dsolve/diffeq/dsol1: general homogeneous successful
dsolve/diffeq/dsol1: > first order, first degree methods :
dsolve/diffeq/dsol1: trying linear bernoulli
dsolve/diffeq/dsol1: trying separable
dsolve/diffeq/dsol1: trying exact
dsolve/diffeq/dsol1: trying general homogeneous
dsolve/diffeq/genhomsol: finding homogeneous solution
dsolve/diffeq/dsol1: general homogeneous successful
/ x \

 2 1/2 2 2 1/2
\(x ) / y(x) (y(x)  (y(x)  4 x ) )
_C1 x exp( 1/4 )
2
x
x = , x = _C1
/ x \

 2 1/2
\(x ) /
2 1/2 2 2 1/2
((x ) y(x) + (y(x)  4 x ) x)
/ x \

 2 1/2
\(x ) /
2 1/2 2 2 1/2
((x ) y(x) + (y(x)  4 x ) x)
/ x \

2 2 1/2  2 1/2
y(x) (y(x) + (y(x)  4 x ) ) / \(x ) /
exp( 1/4 ) / x
2 /
x
> op(simplify(["],power,symbolic));
bytes used=44463800, alloc=6552400, time=46.05
y(x) (y(x)  %1)
_C1 exp( 1/4 )
2
x
x = ,
y(x) + %1
y(x) (y(x) + %1)
x = _C1 (y(x) + %1) exp( 1/4 )
2
x
2 2 1/2
%1 := (y(x)  4 x )
# undet
> dsolve(diff(y(x),x,x)2/x^2*y(x)=7*x^4+3*x^3,y(x));
dsolve/diffeq/polylinearODE: checking Euler equation
dsolve/diffeq/polylinearODE: Euler equation successful
5 _C1 2
y(x) = 1/12 x (2 + 3 x) +  + _C2 x
x
# vector
# changed wrt V.3: solution more complex (wrt constants)
> dsolve({diff(x(t),t)=9*x(t)+2*y(t),diff(y(t),t)=x(t)+8*y(t)},{x(t),y(t)});
dsolve/diffeq/system/linear: determining if system is linear
{x(t) =
1/3 _C1 exp(7 t) + 2/3 _C1 exp(10 t) + 2/3 _C2 exp(10 t)  2/3 _C2 exp(7 t)
, y(t) =
1/3 _C1 exp(10 t)  1/3 _C1 exp(7 t) + 2/3 _C2 exp(7 t) + 1/3 _C2 exp(10 t)
}
# besselJ
> dsolve({x*diff(y(x),x,x)+diff(y(x),x)+2*x*y(x),y(0)=1,D(y)(0)=0},y(x));
dsolve/diffeq/polylinearODE: checking Euler equation
dsolve/diffeq/secorder: checking Bessel's equation
dsolve/diffeq/secorder: Bessel's equation successful
bytes used=46464272, alloc=6552400, time=48.36
1/2
y(x) = BesselJ(0, 2 x)
# separ
# changed wrt V.3: no error any more
> dsolve({x*diff(y(x),x)^2y(x)^2+1,y(0)=1},y(x));
dsolve/diffeq/foxsol: solving high degree d.e. for x
dsolve/diffeq/dsol1: > first order, first degree methods :
dsolve/diffeq/dsol1: trying linear bernoulli
dsolve/diffeq/dsol1: trying separable
dsolve/diffeq/dsol1: trying exact
dsolve/diffeq/inexsol: finding solution to inexact d.e.
dsolve/diffeq/exactsol: finding solution to exact d.e.
dsolve/diffeq/dsol1: exact successful
dsolve/diffeq/dsol1: > first order, first degree methods :
dsolve/diffeq/dsol1: trying linear bernoulli
dsolve/diffeq/dsol1: trying separable
dsolve/diffeq/dsol1: trying exact
dsolve/diffeq/inexsol: finding solution to inexact d.e.
dsolve/diffeq/exactsol: finding solution to exact d.e.
dsolve/diffeq/dsol1: exact successful
bytes used=48467432, alloc=6683448, time=50.84
bytes used=50467984, alloc=7076592, time=53.40
bytes used=52480904, alloc=7207640, time=56.12
bytes used=54503568, alloc=7469736, time=58.97
bytes used=56507992, alloc=7469736, time=61.06
bytes used=58509208, alloc=7469736, time=62.74
bytes used=60513896, alloc=7469736, time=64.71
bytes used=62514168, alloc=7469736, time=66.82
bytes used=64514512, alloc=7469736, time=68.78
bytes used=66515208, alloc=7469736, time=71.04
bytes used=68515824, alloc=7469736, time=73.36
bytes used=70520816, alloc=7469736, time=76.18
bytes used=72536368, alloc=7600784, time=80.47
bytes used=74537264, alloc=7600784, time=83.69
bytes used=76538464, alloc=7600784, time=86.60
bytes used=78545792, alloc=7600784, time=89.23
bytes used=80553616, alloc=7731832, time=92.01
bytes used=82641728, alloc=7731832, time=94.91
bytes used=84693008, alloc=7862880, time=96.66
bytes used=86697496, alloc=7862880, time=98.61
bytes used=88698304, alloc=7862880, time=100.61
bytes used=90698712, alloc=7862880, time=102.59
bytes used=92699224, alloc=7862880, time=104.87
bytes used=94703480, alloc=7862880, time=107.89
bytes used=96704176, alloc=7862880, time=110.66
bytes used=98704504, alloc=7862880, time=114.74
dsolve: Warning: no explicit solutions found
# ic2
> dsolve({diff(y(x),x,x)+y(x)*diff(y(x),x)^3,y(0)=0,D(y)(0)=2},y(x));
dsolve/diffeq/missbody: solving d.e. with missing variable
dsolve/diffeq/dsol1: > first order, first degree methods :
dsolve/diffeq/dsol1: trying linear bernoulli
dsolve/diffeq/bernsol: trying Bernoulli solution
dsolve/diffeq/linearsol: solving 1st order linear d.e.
dsolve/diffeq/dsol1: linear bernoulli successful
bytes used=100707328, alloc=7862880, time=117.84
bytes used=102708264, alloc=7862880, time=120.39
bytes used=104710816, alloc=7862880, time=122.70
bytes used=106711488, alloc=7862880, time=125.04
bytes used=108712928, alloc=7862880, time=127.77
bytes used=110721760, alloc=7862880, time=130.08
2 1/2 1/3 1
y(x) = (3 x + (1 + 9 x ) )  
2 1/2 1/3
(3 x + (1 + 9 x ) )
# SecOrderChangevar
# chaged wrt V.3: no error any more
> eq:=(a*x+b)^2*diff(y(x),x,x)+4*a*(a*x+b)*diff(y(x),x)+2*a^2*y(x):
> dsolve(eq,y(x));
dsolve/diffeq/polylinearODE: checking Euler equation
dsolve/diffeq/secorder: checking Bessel's equation
dsolve/diffeq/secorder: checking hypergeometric equation
dsolve/diffeq/secorder: checking RiemannPapperitz equation
dsolve/diffeq/secorder: trying polynomial solutions to Riccati
dsolve/diffeq/secorder: trying Kovacic's algorithm
dsolve/diffeq/secorder: Kovacic's algorithm successful
_C1 _C2 x
y(x) =  + 
2 2
(a x + b) (a x + b)
# secondord1
> dsolve((x^2x)*diff(w(x),x,x)+(12*x^2)*diff(w(x),x)+(4*x2)*w(x),w(x));
dsolve/diffeq/polylinearODE: checking Euler equation
dsolve/diffeq/secorder: checking Bessel's equation
dsolve/diffeq/secorder: checking hypergeometric equation
dsolve/diffeq/secorder: checking RiemannPapperitz equation
bytes used=112723592, alloc=7862880, time=132.46
dsolve/diffeq/secorder: trying polynomial solutions to Riccati
dsolve/diffeq/secorder: polynomial solutions to Riccati successful
2
w(x) = _C1 exp(2 x) + _C2 x
# variation
# changed wrt V.3: output simpler
> dsolve(diff(y(x),x,x)+y(x)=csc(x),y(x));
dsolve/diffeq/polylinearODE: trying linear constant coefficient
dsolve/diffeq/polylinearODE: linear constant coefficient successful
y(x) = ln(sin(x)) sin(x)  x cos(x) + _C1 sin(x) + _C2 cos(x)
# triangular
# changed wrt V.3: now expressed in terms of DESol of 2nd order
> dsolve({D(x)(t)=x(t)*(1+cos(t)/(2+sin(t))),D(y)(t)=x(t)y(t)},{x(t),y(t)});
bytes used=114723920, alloc=7862880, time=134.76
dsolve/diffeq/system/linear: determining if system is linear
dsolve/diffeq/linearODE: checking Bessel's equation
/d \
{x(t) = %2 _C1 + _C1  %2, y(t) = %2 _C1}
\dt /
2
d
%1 :=  _Y(t)
2
dt
/ /d \
%2 := DESol({ 2 %1  %1 sin(t) + cos(t)  _Y(t) + 2 _Y(t) + _Y(t) sin(t)
\ \dt /
\
+ _Y(t) cos(t)/(2 + sin(t))}, {_Y(t)})
/
# highOrder
# changed wrt V.3: now produces an error !!!
> dsolve( {
> diff(x(t),t)x(t)+2*y(t)=0,diff(x(t),t$2)2*diff(y(t),t)=2*tcos(2*t)},
> {x(t),y(t)} );
dsolve/diffeq/system/linear: determining if system is linear
Error, (in dsolve/diffeq/ConvertSysTo1stOrder)
unable to convert to an explicit firstorder system
# inhomo
# changed wrt V.3: now finds the solution
> eq:= {diff(y1(x),x)=1/(x*(x^2+1))*y1(x)+1/(x^2*(x^2+1))*y2(x)+1/x,
> diff(y2(x),x)=x^2/(x^2+1)*y1(x) + (2*x^2+1)/(x*(x^2+1))*y2(x)+1}:
> dsolve( eq,{y1(x),y2(x)} );
dsolve/diffeq/system/linear: determining if system is linear
bytes used=116724264, alloc=7862880, time=137.02
dsolve/diffeq/polylinearODE: checking Euler equation
dsolve/diffeq/polylinearODE: Euler equation successful
x ln(x)  x + _C1 x + _C2 2
{y1(x) = , y2(x) = x ln(x)  x + _C1 x  _C2 x }
x
# transfBernoulli
> dsolve(3*diff(y(x),x,x)*diff(y(x),x$4)5*diff(y(x),x$3)^2=0,y(x));
dsolve/diffeq/missbody: solving d.e. with missing variable
dsolve/diffeq/missbody: solving d.e. with missing variable
dsolve/diffeq/missbody: solving d.e. with missing variable
dsolve/diffeq/dsol1: > first order, first degree methods :
dsolve/diffeq/dsol1: trying linear bernoulli
dsolve/diffeq/linearsol: solving 1st order linear d.e.
dsolve/diffeq/dsol1: linear bernoulli successful
bytes used=118724984, alloc=7862880, time=139.37
1/2
(6 _C1 x  6 _C2 _C1)
y(x) = 3  + _C3 x + _C4,
2
_C1
1/2
(6 _C1 x  6 _C2 _C1)
y(x) = 3  + _C3 x + _C4
2
_C1
# nthorder
> dsolve({diff(y(x),x$4)=sin(x),y(0)=0,D(y)(0)=0,(D@@2)(y)(0)=0,(D@@3)(y)(0)=0},y(x));
dsolve/diffeq/polylinearODE: trying linear constant coefficient
dsolve/diffeq/polylinearODE: linear constant coefficient successful
3
y(x) = sin(x)  x + 1/6 x
# bronstein
> a0:=104/25*x^10+(274/2522/15*sqrt(222))*x^8+(7754/7568/15*sqrt(222))*x^6
> +(11248/75194/15*sqrt(222))*x^4+(29452/75296/5*sqrt(222))*x^2
> 10952/5148/3*sqrt(222):
> a2:=x^12+2*x^10+151/3*x^8+296/3*x^6+5920/9*x^4+10952/9*x^2+5476/9:
> eq:=a2*diff(y(x),x,x)a0*y(x);
12 10 8 6 4 2
eq := (x + 2 x + 151/3 x + 296/3 x + 5920/9 x + 10952/9 x + 5476/9)
/ 2 \
d  /104 10 /274 22 1/2\ 8 /7754 68 1/2\ 6
 y(x)   x +    I 222  x +    I 222  x
 2  \25 \25 15 / \ 75 15 /
\dx /
/11248 194 1/2\ 4 /29452 1/2\ 2
+    I 222  x +   296/5 I 222  x  10952/5
\ 75 15 / \ 75 /
1/2\
 148/3 I 222  y(x)
/
> dsolve(eq,y(x));
bytes used=120725320, alloc=7862880, time=141.80
bytes used=122725824, alloc=7862880, time=143.94
dsolve/diffeq/polylinearODE: checking Euler equation
dsolve/diffeq/secorder: checking Bessel's equation
dsolve/diffeq/secorder: checking hypergeometric equation
dsolve/diffeq/secorder: checking RiemannPapperitz equation
bytes used=124726344, alloc=7862880, time=146.07
bytes used=126733544, alloc=7862880, time=148.17
bytes used=128735432, alloc=7993928, time=150.04
bytes used=130736632, alloc=7993928, time=151.91
bytes used=132737184, alloc=7993928, time=153.87
bytes used=134745760, alloc=7993928, time=155.69
bytes used=136754456, alloc=8124976, time=157.46
bytes used=138756016, alloc=8124976, time=159.32
bytes used=140765520, alloc=8124976, time=161.11
bytes used=142779352, alloc=8124976, time=162.82
bytes used=144784848, alloc=8124976, time=164.82
bytes used=146786240, alloc=8124976, time=166.61
bytes used=148786728, alloc=8124976, time=168.52
bytes used=150794496, alloc=8124976, time=170.42
bytes used=152796144, alloc=8124976, time=172.24
bytes used=154798208, alloc=8124976, time=174.06
dsolve/diffeq/secorder: trying polynomial solutions to Riccati
bytes used=156798536, alloc=8124976, time=175.63
dsolve/diffeq/secorder: trying Kovacic's algorithm
bytes used=158801512, alloc=8124976, time=177.97
bytes used=160803480, alloc=8124976, time=180.28
bytes used=162805176, alloc=8124976, time=182.62
bytes used=164805744, alloc=8124976, time=185.01
bytes used=166806448, alloc=8124976, time=187.40
. . .
. . .
bytes used=1985551784, alloc=8124976, time=2377.91
bytes used=1987555448, alloc=8124976, time=2380.32
bytes used=1989555840, alloc=8124976, time=2382.76
>>>>>> Had to stop it here.
> # moussiaux
> dsolve(15*diff(y(x),x)+24*y(x)^2=7*x^(8/3),y(x));
dsolve/diffeq/dsol1: > first order, first degree methods :
dsolve/diffeq/dsol1: trying linear bernoulli
dsolve/diffeq/dsol1: trying separable
dsolve/diffeq/dsol1: trying exact
dsolve/diffeq/dsol1: trying general homogeneous
dsolve/diffeq/dsol1: trying Riccati
dsolve: Warning: no solutions found
> # labahn1
> dsolve((x1)*diff(y(x),x,x)+(3/2x)*diff(y(x),x)+y(x)/2,y(x));
dsolve/diffeq/polylinearODE: checking Euler equation
dsolve/diffeq/secorder: checking Bessel's equation
dsolve/diffeq/secorder: checking hypergeometric equation
dsolve/diffeq/secorder: hypergeometric equation successful
bytes used=2000112, alloc=1703624, time=0.97
1/2
y(x) = _C1 (x  1) + _C2 hypergeom([1/2], [1/2], x  1)
> # labahn2
> dsolve(diff(y(x),x,x)(x^62*x^5+3*x^4+x^3+7/4*x^25*x+1)/x^4*y(x),y(x));
dsolve/diffeq/polylinearODE: checking Euler equation
dsolve/diffeq/secorder: checking Bessel's equation
dsolve/diffeq/secorder: checking hypergeometric equation
dsolve/diffeq/secorder: checking RiemannPapperitz equation
bytes used=4000776, alloc=2752008, time=2.01
dsolve/diffeq/secorder: trying polynomial solutions to Riccati
dsolve/diffeq/secorder: trying Kovacic's algorithm
bytes used=6001488, alloc=3800392, time=3.66
bytes used=8001968, alloc=4193536, time=5.67
dsolve/diffeq/secorder: Kovacic's algorithm successful
bytes used=10099904, alloc=4717728, time=7.51
/

y(x) = _C1 exp(1/2 %1) + _C2 exp(1/2 %1)  exp(%1) dx

/
2 3
2 x + x  2  3 ln(x) x + 2 ln(x  1) x + 2 ln(1 + x) x
%1 := 
x
> # labahn3
dsolve(diff(y(x),x,x)+(x^4+1)*y(x),y(x));
> dsolve(diff(y(x),x,x)+(x^4+1)*y(x),y(x));
dsolve/diffeq/polylinearODE: checking Euler equation
dsolve/diffeq/secorder: checking Bessel's equation
dsolve/diffeq/secorder: checking hypergeometric equation
dsolve/diffeq/secorder: checking RiemannPapperitz equation
dsolve/diffeq/secorder: trying polynomial solutions to Riccati
bytes used=12100376, alloc=5110872, time=9.91
dsolve/diffeq/secorder: trying Kovacic's algorithm
dsolve/diffeq/expsols: trying exponential solutions
dsolve/diffeq/expsols_solvericcati: max # of trials : 0
/ 2 \
d  4
y(x) = DESol({ _Y(x) + (x + 1) _Y(x)}, {_Y(x)})
 2 
\dx /
> # labahn4
> dsolve(diff(y(x),x$3)+x*diff(y(x),x)+y(x),y(x));
dsolve/diffeq/polylinearODE: checking Euler equation
dsolve/diffeq/expsols: trying exponential solutions
dsolve/diffeq/expsols_solvericcati: max # of trials : 0
/ 3 \
d  /d \
y(x) = DESol({ _Y(x) + x  _Y(x) + _Y(x)}, {_Y(x)})
 3  \dx /
\dx /
> # labahn5
> dsolve((6+8*x^2)*y(x)+(11+4*x12*x^2)*diff(y(x),x)+(66*x+4*x^2)*diff(y(x),x$2
> )
> +(1+2*x)*diff(y(x),x$3),y(x));
dsolve/diffeq/polylinearODE: checking Euler equation
dsolve/diffeq/expsols: trying exponential solutions
bytes used=14101216, alloc=5110872, time=12.22
dsolve/diffeq/expsols_padicpartbounded: eqns 1 deg 3
dsolve/diffeq/expsols_solvericcati: max # of trials : 0
dsolve/diffeq/expsols: expon. solutions partially successful. Result(s) = exp(Int(1,
x)), exp(Int(2,x))
dsolve/diffeq/expsols_reduceorder: reduction to order 2
bytes used=16102048, alloc=5110872, time=14.86
dsolve/diffeq/expsols_reduceorder: reduction to order 1
dsolve/diffeq/expsols_reduceorder: back in order 2
dsolve/diffeq/expsols_reduceorder: back in order 3
y(x) = _C1 exp(x) + _C2 exp(2 x) + _C3 exp(x) erf(x)
> # labahn6
> dsolve((3+6*x+20*x^240*x^3+16*x^432*x^5)*y(x)+(315*x+44*x^3+48*x^5)*D(y)(x)
> +(9*x26*x^224*x^416*x^5)*diff(y(x),x,x)+(6*x^24*x^3+8*x^4)*diff(y(x),x$3),
> y(x));
dsolve/diffeq/polylinearODE: checking Euler equation
dsolve/diffeq/expsols: trying exponential solutions
bytes used=18104352, alloc=5241920, time=17.25
bytes used=20105112, alloc=5372968, time=19.71
dsolve/diffeq/expsols_padicpartbounded: eqns 2 deg 3
bytes used=22105432, alloc=5372968, time=22.18
dsolve/diffeq/expsols_padicpartbounded: eqns 1 deg 3
dsolve/diffeq/expsols_solvericcati: max # of trials : 35
bytes used=24105776, alloc=5372968, time=24.39
bytes used=26106328, alloc=5504016, time=26.81
bytes used=28106880, alloc=5504016, time=29.37
dsolve/diffeq/expsols: expon. solutions partially successful. Result(s) = exp(Int(1,
x))
dsolve/diffeq/expsols_reduceorder: reduction to order 2
dsolve/diffeq/polylinearODE: checking Euler equation
dsolve/diffeq/secorder: checking Bessel's equation
dsolve/diffeq/secorder: checking hypergeometric equation
dsolve/diffeq/secorder: checking RiemannPapperitz equation
bytes used=30107224, alloc=5766112, time=31.57
bytes used=32111896, alloc=5766112, time=33.41
bytes used=34112280, alloc=5897160, time=35.03
bytes used=36118112, alloc=5897160, time=36.80
bytes used=38119552, alloc=6028208, time=38.46
dsolve/diffeq/secorder: trying polynomial solutions to Riccati
dsolve/diffeq/secorder: trying Kovacic's algorithm
bytes used=40120024, alloc=6028208, time=40.57
bytes used=42120936, alloc=6028208, time=42.90
dsolve/diffeq/expsols: trying exponential solutions
bytes used=44121640, alloc=6028208, time=45.25
bytes used=46121984, alloc=6028208, time=47.77
dsolve/diffeq/expsols_padicpartbounded: eqns 2 deg 2
dsolve/diffeq/expsols_padicpartbounded: eqns 1 deg 2
dsolve/diffeq/expsols_solvericcati: max # of trials : 9
bytes used=48123328, alloc=6028208, time=50.20
dsolve/diffeq/expsols: exponential solutions successful
dsolve/diffeq/expsols_reduceorder: back in order 3
/
 2 1/2
y(x) = _C1 exp(x) + _C2 exp(x) Ei(1, x) + _C3 exp(x)  exp(x ) x dx

/
> # labahn7
> dsolve(diff(y(x),x$3)+x*diff(y(x),x)+y(x)=(5+2*x+x^2)/(x+1)^4,y(x));
dsolve/diffeq/polylinearODE: checking Euler equation
dsolve/diffeq/expsols: trying exponential solutions
dsolve/diffeq/expsols_solvericcati: max # of trials : 0
/ 3 \
1 d  /d \
y(x) =  + DESol({ _Y(x) + x  _Y(x) + _Y(x)}, {_Y(x)})
1 + x  3  \dx /
\dx /
bytes used=50133776, alloc=6028208, time=52.54
> # labahn8
> dsolve(diff(y(x),x$3)+x*diff(y(x),x,x)+(x^2+3)*diff(y(x),x)+(x^3+x)*y(x),y(x));
dsolve/diffeq/polylinearODE: checking Euler equation
dsolve/diffeq/expsols: trying exponential solutions
dsolve/diffeq/expsols_solvericcati: max # of trials : 0
dsolve/diffeq/expsols: expon. solutions partially successful. Result(s) = exp(Int(x
,x))
dsolve/diffeq/expsols_reduceorder: reduction to order 2
dsolve/diffeq/polylinearODE: checking Euler equation
dsolve/diffeq/secorder: checking Bessel's equation
dsolve/diffeq/secorder: checking hypergeometric equation
dsolve/diffeq/secorder: checking RiemannPapperitz equation
dsolve/diffeq/secorder: trying polynomial solutions to Riccati
dsolve/diffeq/secorder: trying Kovacic's algorithm
bytes used=52134168, alloc=6028208, time=54.89
dsolve/diffeq/expsols: trying exponential solutions
dsolve/diffeq/expsols_solvericcati: max # of trials : 0
dsolve/diffeq/expsols_reduceorder: back in order 3
2
y(x) = _C1 exp( 1/2 x )
/ / 2 \
2  d  /d \ 2
+ exp( 1/2 x )  DESol({ _Y(x)  2 x  _Y(x) + 2 x _Y(x)}, {_Y(x)}) dx
  2  \dx /
/ \dx /
> # labahn9
> dsolve(diff(y(x),x$5)+2*diff(y(x),x)+2*y(x),y(x));
dsolve/diffeq/polylinearODE: trying linear constant coefficient
dsolve/diffeq/polylinearODE: linear constant coefficient successful

\
y(x) = ) _C1[_R] exp(_R x)
/

_R = %1
5
%1 := RootOf(2 + 2 _Z + _Z )
> # labahn10
> dsolve(diff(y(x),x$5)+4*diff(y(x),x$3)+4*diff(y(x),x$2)+4*D(y)(x)+8*y(x),y(x));
dsolve/diffeq/polylinearODE: trying linear constant coefficient
dsolve/diffeq/polylinearODE: linear constant coefficient successful
bytes used=54134536, alloc=6028208, time=57.24
bytes used=56135120, alloc=6028208, time=59.42
bytes used=58137016, alloc=6290304, time=61.52
bytes used=60137448, alloc=6290304, time=63.65
bytes used=62138272, alloc=6290304, time=65.80
1/2 1/2
y(x) = _C1 sin(2 x) + _C2 cos(2 x) + _C3
1/2 1/3 1/2 1/3 1/2 1/3 1/2
exp(1/18 (54 + 6 87 ) (6  9 (54 + 6 87 ) + (54 + 6 87 ) 87 ) x)
1/3
+ _C4 exp( 1/36 %1
1/3 1/2 1/2 1/3 1/2 1/3 1/2 1/3 1/2
(6 + %1 29 3  9 %1  6 I 3  3 I %1 29 + 9 I %1 3 ) x)
1/3
+ _C5 exp( 1/36 %1
1/3 1/2 1/2 1/3 1/2 1/3 1/2 1/3 1/2
(6 + %1 29 3  9 %1 + 6 I 3 + 3 I %1 29  9 I %1 3 ) x)
1/2 1/2
%1 := 54 + 6 29 3
> # labahn11
> dsolve(x^6*diff(y(x),x$6)+15*x^5*diff(y(x),x$5)+69*x^4*diff(y(x),x$4)
> +118*x^3*diff(y(x),x$3)+75*x^2*diff(y(x),x$2)+21*x*diff(y(x),x)+4*y(x),y(x));
dsolve/diffeq/polylinearODE: checking Euler equation
dsolve/diffeq/polylinearODE: Euler equation successful
/  \ /  \
 \ _R  \ _R 
y(x) =  ) _C1[_R] x  +  ) _C2[_R] x ln(x)
 /   / 
     
\_R = %1 / \_R = %1 /
3
%1 := RootOf(_Z + 2 _Z + 2)
> # labahn12
> ode:=(2*x^2+x+n^2)*y(x)+(4*x^22*xn^2)*diff(y(x),x)
> +(3*x^2+x)*diff(y(x),x$2)+x^2*diff(y(x),x$3):
> dsolve(ode,y(x));
dsolve/diffeq/polylinearODE: checking Euler equation
dsolve/diffeq/expsols: trying exponential solutions
bytes used=64138800, alloc=6290304, time=68.12
bytes used=66140136, alloc=6290304, time=70.51
dsolve/diffeq/expsols_padicpartbounded: eqns 1 deg 3
dsolve/diffeq/expsols_solvericcati: max # of trials : 8
bytes used=68140512, alloc=6290304, time=73.30
dsolve/diffeq/expsols: expon. solutions partially successful. Result(s) = exp(Int(1,
x))
dsolve/diffeq/expsols_reduceorder: reduction to order 2
dsolve/diffeq/polylinearODE: checking Euler equation
dsolve/diffeq/secorder: checking Bessel's equation
dsolve/diffeq/secorder: Bessel's equation successful
dsolve/diffeq/expsols_reduceorder: back in order 3
bytes used=70141504, alloc=6421352, time=75.63
/ /
 
y(x) = _C1 exp(x) + _C2 exp(x)  BesselJ(n, x) dx + _C3 exp(x)  BesselY(n, x) dx
 
/ /
> # labahn13
> dsolve(subs(n=1,ode),y(x));
dsolve/diffeq/polylinearODE: checking Euler equation
dsolve/diffeq/expsols: trying exponential solutions
bytes used=72142592, alloc=6421352, time=78.09
dsolve/diffeq/expsols_padicpartbounded: eqns 1 deg 3
dsolve/diffeq/expsols_solvericcati: max # of trials : 0
dsolve/diffeq/expsols: expon. solutions partially successful. Result(s) = exp(Int(1,
x))
dsolve/diffeq/expsols_reduceorder: reduction to order 2
dsolve/diffeq/polylinearODE: checking Euler equation
dsolve/diffeq/secorder: checking Bessel's equation
dsolve/diffeq/secorder: Bessel's equation successful
dsolve/diffeq/expsols_reduceorder: back in order 3
y(x) = _C1 exp(x) + _C2 exp(x) BesselY(0, x) + _C3 exp(x) BesselJ(0, x)
> # labahn14
> dsolve(diff(y(x),x$2)+3/x*diff(y(x),x)+(x^2143)/x^2*y(x)=x140/x,y(x));
dsolve/diffeq/polylinearODE: checking Euler equation
dsolve/diffeq/secorder: checking Bessel's equation
dsolve/diffeq/secorder: Bessel's equation successful
_C1 BesselY(12, x) _C2 BesselJ(12, x)
y(x) = x +  + 
x x
> # labahn15
> dsolve(diff(y(x),x)+x*y(x)^2=1,y(x));
dsolve/diffeq/dsol1: > first order, first degree methods :
dsolve/diffeq/dsol1: trying linear bernoulli
dsolve/diffeq/dsol1: trying separable
dsolve/diffeq/dsol1: trying exact
dsolve/diffeq/dsol1: trying general homogeneous
dsolve/diffeq/dsol1: trying Riccati
dsolve/diffeq/polylinearODE: checking Euler equation
dsolve/diffeq/secorder: checking Bessel's equation
dsolve/diffeq/secorder: Bessel's equation successful
bytes used=74143032, alloc=6421352, time=80.58
dsolve/diffeq/dsol1: Riccati successful
3/2 3/2
_C1 BesselK(1/3, 2/3 x )  BesselI(1/3, 2/3 x )
y(x) =  
1/2 3/2 3/2
x (_C1 BesselK(2/3, 2/3 x ) + BesselI(2/3, 2/3 x ))