|\^/| Maple V Release 4 (WM - Internal Use Only) ._|\| |/|_. Copyright (c) 1981-1996 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^4-x^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)^2-2*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^4-2*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^2-x)*diff(u(x),x,x)+(2*x^2+4*x-3)*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 Riemann-Papperitz 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)^2-1+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^2-1)*diff(y(x),x)^2-2*x*y(x)*diff(y(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 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(t-1)=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 Riemann-Papperitz 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((1-x)*(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^2-y(x)^2-7)/(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 non-linear systems dsolve/diffeq/system/linear: determining if system is linear dsolve/diffeq/system: cannot solve non-linear systems dsolve: Warning: no solutions found # liouvillian > dsolve((x^3/2-x^2)*diff(y(x),x,x)+(2*x^2-3*x+1)*diff(y(x),x) > +(x-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 Riemann-Papperitz 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*x-2)/(x^3-2*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 Riemann-Papperitz 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)^2-y(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)^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 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)^2-x(t)^2)-2*b(t)*x(t)*y(t)-2*c*x(t), > diff(y(t),t)-b(t)*(y(t)^2-x(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 non-linear systems dsolve/diffeq/system/linear: determining if system is linear dsolve/diffeq/system: cannot solve non-linear 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)^2-y(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(x-y(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)^2-y(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 Riemann-Papperitz 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^2-x)*diff(w(x),x,x)+(1-2*x^2)*diff(w(x),x)+(4*x-2)*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 Riemann-Papperitz 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*t-cos(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 first-order 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+(274/25-22/15*sqrt(-222))*x^8+(7754/75-68/15*sqrt(-222))*x^6 > +(11248/75-194/15*sqrt(-222))*x^4+(29452/75-296/5*sqrt(-222))*x^2 > -10952/5-148/3*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 Riemann-Papperitz 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((x-1)*diff(y(x),x,x)+(3/2-x)*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^6-2*x^5+3*x^4+x^3+7/4*x^2-5*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 Riemann-Papperitz 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 Riemann-Papperitz 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*x-12*x^2)*diff(y(x),x)+(-6-6*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^2-40*x^3+16*x^4-32*x^5)*y(x)+(-3-15*x+44*x^3+48*x^5)*D(y)(x) > +(9*x-26*x^2-24*x^4-16*x^5)*diff(y(x),x,x)+(6*x^2-4*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 Riemann-Papperitz 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 Riemann-Papperitz 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^2-2*x-n^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^2-143)/x^2*y(x)=x-140/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 ))