# input/output file for Maple V.3 infolevel[dsolve]:=10: # firstord1 dsolve((x^4-x^3)*diff(u(x),x) + 2*x^4*u(x) = x^3/3 + C,u(x)); 3 2 2 (1/6 exp(2 x) x - 1/4 exp(2 x) x + 1/2 exp(2 x) C + _C1 x ) exp(- 2 x) u(x) = ------------------------------------------------------------------------ 2 2 (x - 2 x + 1) x # firstord2 dsolve(-1/2*diff(u(x),x)+u(x)=sin(x),u(x)); 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)); 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/dsol1: general homogeneous successful 2 2 1/2 (x + 2 y(x) ) x = _C1 x exp(- 1/3 x (arctanh(-----------------) 2 1/2 (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 # it works with a rational n dsolve(diff(y(x),x)+P(x)*y(x)=Q(x)*y(x)^(2/3),y(x)); / / / 1/3 | | | y(x) = 1/3 exp(- 1/3 | P(x) dx) | exp(1/3 | P(x) dx) Q(x) dx | | | / / / / | + exp(- 1/3 | P(x) dx) _C1 | / # 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 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/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/secorder: Kovacic's algorithm successful - 3 + 2 x _C1 exp(- 2 x) _C2 u(x) = 1/12 --------- + -------------- + ----------- 2 2 2 2 (x - 1) (x - 1) (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/missbody: solving d.e. with missing variable dsolve/diffeq/dsol1: -> first order, first degree methods : 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 y(x) / 1/2 | 6 x = | 1/2 ------------------------------------ dy1 - _C2, | / 2 1 2 \1/2 / |3 y1 ln(y1) + ---- + 3 y1 _C1| 0 \ y1 / y(x) / 1/2 | 6 x = | - 1/2 ------------------------------------ dy2 - _C2 | / 2 1 2 \1/2 / |3 y2 ln(y2) + ---- + 3 y2 _C1| 0 \ y2 / # clairaut dsolve((x^2-1)*diff(y(x),x)^2-2*x*y(x)*diff(y(x),x)+y(x)^2-1,y(x)); 2 1/2 2 1/2 y(x) = x _C1 - (_C1 + 1) , y(x) = x _C1 + (_C1 + 1) , / 2 1/2\ / 2 1/2\ | (- y(x) + 1) | | (- y(x) + 1) | x = |x = - ----------------|, x = |x = ----------------| | / 1 \1/2 | | / 1 \1/2 | | |-----| y(x)| | |-----| y(x)| | | 2| | | | 2| | \ \y(x) / / \ \y(x) / / # clairaut2 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=5009676, alloc=2031244, time=13.00 Error, (in solve) To solve differential equations, please use dsolve # 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 dsolve(diff(y(t),t)+a*y(t-1)=0,y(t)); dsolve/diffeq/dsol1: -> first order, first degree methods : 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 dsolve(diff(y(x,a),x)=a*y(x,a),y(x,a)); Error, (in dsolve) PDEs not handled yet # 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: 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: 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: Euler equation successful _C2 2 4 y(x) = _C1 x + --- + _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: trying exact 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 # Nonlinear, 3th order 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 2 2 2 2 2 1/2 I (x _C1 + 2 x _C1 _C2 + _C2 _C1 - 1) y(x) = --------------------------------------------- + _C3 _C1 # Nonlinear, 4th order dsolve( diff(y(x),x\$2)*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. bytes used=5001432, alloc=2031244, time=14.88 dsolve/diffeq/dsol1: linear bernoulli successful bytes used=6001604, alloc=2031244, time=18.67 bytes used=7001764, alloc=2096768, time=23.00 bytes used=8002096, alloc=2096768, time=27.38 bytes used=9002404, alloc=2096768, time=31.57 bytes used=10002704, alloc=2096768, time=35.57 bytes used=11002896, alloc=2096768, time=39.50 bytes used=12003040, alloc=2096768, time=43.68 bytes used=13003204, alloc=2096768, time=47.90 bytes used=14003340, alloc=2096768, time=52.12 bytes used=15003564, alloc=2096768, time=55.93 bytes used=16003972, alloc=2096768, time=60.15 bytes used=17004152, alloc=2096768, time=64.38 bytes used=18004464, alloc=2096768, time=68.62 bytes used=19004656, alloc=2096768, time=72.90 bytes used=20004828, alloc=2096768, time=76.78 bytes used=21005024, alloc=2096768, time=81.02 bytes used=22005268, alloc=2096768, time=85.25 bytes used=23005628, alloc=2096768, time=89.52 y(x) = 4/21 %9 + 4/21 %8 - 4/21 %7 + 4/21 %6 + _C3 x - 4/21 %5 + 4/21 %4 - 4/21 %3 - 4/21 %2 + _C4, y(x) = - 4/21 %9 - 4/21 %8 + 4/21 %7 - 4/21 %6 + _C3 x + 4/21 %5 - 4/21 %4 + 4/21 %3 + 4/21 %2 + _C4, y(x) = - 4/21 %9 - 4/21 %8 - 4/21 %7 + 4/21 %6 + _C3 x + 4/21 %5 - 4/21 %4 - 4/21 %3 - 4/21 %2 + _C4, y(x) = 4/21 %9 + 4/21 %8 + 4/21 %7 - 4/21 %6 + _C3 x - 4/21 %5 + 4/21 %4 + 4/21 %3 + 4/21 %2 + _C4 %1 := 1/8 Pi signum(_C1) 7/4 1/2 1/2 I (x + _C2) (2 - 2 ) sin(%1) %2 := ------------------------------------ 1/4 abs(_C1) 7/4 1/2 1/2 I (x + _C2) (2 + 2 ) cos(%1) %3 := ------------------------------------ 1/4 abs(_C1) 7/4 1/2 1/2 I (x + _C2) (2 + 2 ) sin(%1) %4 := ------------------------------------ 1/4 abs(_C1) 7/4 1/2 1/2 I (x + _C2) (2 - 2 ) cos(%1) %5 := ------------------------------------ 1/4 abs(_C1) 7/4 1/2 1/2 (x + _C2) (2 + 2 ) sin(%1) %6 := ---------------------------------- 1/4 abs(_C1) 7/4 1/2 1/2 (x + _C2) (2 - 2 ) cos(%1) %7 := ---------------------------------- 1/4 abs(_C1) 7/4 1/2 1/2 (x + _C2) (2 - 2 ) sin(%1) %8 := ---------------------------------- 1/4 abs(_C1) 7/4 1/2 1/2 (x + _C2) (2 + 2 ) cos(%1) %9 := ---------------------------------- 1/4 abs(_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) = 1/2 x + _C1, y(x) = - x + 1 + exp(- 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 constant-coefficient 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)); / | 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) ) - 3/4 exp(- 2/3 y(x) ) - 1/2 exp(- 2/3 y(x) ) 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 / 1/3 x \2 |- 1/2 %1 - 1/3 ------- - 1/2 %2| x | 1/3 | \ a %1 / / 1/3 x \4 - 3/4 a |- 1/2 %1 - 1/3 ------- - 1/2 %2| = _C1, ..., ... | 1/3 | \ a %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: trying polynomial solutions to Riccati dsolve/diffeq/secorder: trying Kovacic's algorithm dsolve/diffeq/secorder: Kovacic's algorithm successful bytes used=8024436, alloc=2162292, time=23.80 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 bytes used=9024580, alloc=2293340, time=27.32 dsolve/diffeq/dsol1: Riccati successful (- _C1 cos(x) + sin(x)) exp(- x) y(x) = -------------------------------- _C1 sin(x) + cos(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 constant-coefficient dsolve: Warning: no solutions found # scaleinv dsolve(x^2*diff(y(x),x,x)+3*x*diff(y(x),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 1/3 y(x) + 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/dsol1: general homogeneous successful 2 2 1/2 2 1/2 (x + y(x) ) (x ) arctanh(---------------) 2 1/2 (x ) _C1 x exp(--------------------------------) x x = -------------------------------------------, y(x) 2 2 1/2 2 1/2 (x + y(x) ) (x ) arctanh(---------------) 2 1/2 (x ) _C1 x exp(- --------------------------------) x x = --------------------------------------------- y(x) # solvabley dsolve(x-y(x)*diff(y(x),x)+x*diff(y(x),x)^2,y(x)); 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 \ |-------| | 2 1/2| 2 1/2 2 2 1/2 \(x ) / ((x ) y(x) + x (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 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 constant-coefficient dsolve/diffeq/linsys: solving linear system {y(t) = _C1 exp(10 t) + _C2 exp(7 t), x(t) = 2 _C1 exp(10 t) - _C2 exp(7 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 1/2 y(x) = BesselJ(0, 2 x) # separ dsolve({x*diff(y(x),x)^2-y(x)^2+1,y(0)=1},y(x)); bytes used=20025432, alloc=4062488, time=12.30 Error, (in solve/sumint) cannot solve for variables used in unevaluated sum/ints # ic2 dsolve({diff(y(x),x,x)+y(x)*diff(y(x),x)^3,y(0)=0,D(y)(0)=2},y(x)); 2 1/2 1/3 1 y(x) = (3 x + (1 + 9 x ) ) - ------------------------ 2 1/2 1/3 (3 x + (1 + 9 x ) ) # SecOrderChangevar 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)); Error, (in dsolve/diffeq/bessel) unable to compute coeff # 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: trying polynomial solutions to Riccati dsolve/diffeq/secorder: polynomial solutions to Riccati successful 2 w(x) = _C1 exp(2 x) + _C2 x # variation 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 sin(x) y(x) = - 2 arctan(----------) cos(x) + sin(x) ln(sin(x)) - sin(x) ln(2) cos(x) + 1 + _C1 cos(x) + _C2 sin(x) # triangular dsolve({D(x)(t)=x(t)*(1+cos(t)/(2+sin(t))),D(y)(t)=x(t)-y(t)},{x(t),y(t)}); dsolve/diffeq/system/linear: determining if system is linear constant-coefficient dsolve: Warning: no solutions found # highOrder > 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)} ); { x(t) = 2 - t + 1/34 sin(2 t) - 4 t - 9 + 2/17 cos(2 t) + 2 _C1 + 4 _C2 exp(1/2 t ), 2 2 y(t) = 1/2 cos(t) sin(t) - t - cos(t) - 1/2 t + 9/17 cos(2 t) - 2/17 sin(2 t) - 2 + _C1 + _C2 exp(1/2 t) } # inhomo > 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}; 2 2 d x y1(x) (2 x + 1) y2(x) eq := {---- y2(x) = - -------- + ---------------- + 1, dx 2 2 x + 1 x (x + 1) d y1(x) y2(x) ---- y1(x) = - ---------- + ----------- + 1/x} dx 2 2 2 x (x + 1) x (x + 1) > dsolve( eq,{y1(x),y2(x)} ); # circle dsolve((diff(y(x),x)^2+1)*diff(y(x),x\$3)-3*diff(y(x),x)*diff(y(x),x,x)^2,y(x)); 2 2 2 2 2 1/2 I (x _C1 + 2 x _C1 _C2 + _C2 _C1 - 1) y(x) = --------------------------------------------- + _C3 _C1 # transfBernoulli dsolve(3*diff(y(x),x,x)*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 # 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: 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); dsolve(eq,y(x)); bytes used=7450436148, alloc=35907152, time=11569.53 Error, (in expand/bigprod) object too large # check the solution sol:=x*(3*x^2 + sqrt(-222))*(x^2+1)^(-3/5)*(x^4+74/3)^(1/5): expand(numer(eval(subs(y(x)=sol,eq)))); 0 # moussiaux dsolve(15*diff(y(x),x)+24*y(x)^2=7*x^(-8/3),y(x)); dsolve: Warning: no solutions found sol:=(56*x^(2/3)*exp(18*sqrt(14)/5/x^(1/3)/sqrt(3))*sqrt(3) > +25*x^(1/3)*exp(18*sqrt(14)/5/x^(1/3)/sqrt(3))*sqrt(3)*x > -30*exp(18*sqrt(14)/5/x^(1/3)/sqrt(3))*sqrt(14)*x > +56*x^(2/3)*exp(6*sqrt(14)/5/x^(1/3)/sqrt(3))*sqrt(3)*K > +25*x^(1/3)*exp(6*sqrt(14)/5/x^(1/3)/sqrt(3))*sqrt(3)*K*x > +30*exp(6*sqrt(14)/5/x^(1/3)/sqrt(3))*sqrt(14)*K*x)/ > (40*x^(1/3)*exp(18*sqrt(14)/5/x^(1/3)/sqrt(3))*sqrt(3)*x^2 > -48*exp(18*sqrt(14)/5/x^(1/3)/sqrt(3))*sqrt(14)*x^2 > +40*x^(1/3)*exp(6*sqrt(14)/5/x^(1/3)/sqrt(3))*sqrt(3)*K*x^2 > +48*exp(6*sqrt(14)/5/x^(1/3)/sqrt(3))*sqrt(14)*K*x^2); # labahn1 dsolve((x-1)*diff(y(x),x,x)+(3/2-x)*diff(y(x),x)+y(x)/2,y(x)); 1/2 y(x) = _C1 hypergeom([-1/2], [1/2], x - 1) + _C2 (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)); / | y(x) = _C1 exp(1/2 %1) + _C2 exp(1/2 %1) | exp(- %1) dx | / 3 2 x - 2 x - 2 - 3 ln(x) x + 2 ln(x - 1) x + 2 ln(x + 1) x %1 := --------------------------------------------------------- x # labahn3 dsolve(diff(y(x),x,x)+(x^4+1)*y(x),y(x)); / 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)); / 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)); 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)); / | 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)); / 3 \ 1 | d | / d \ y(x) = ----- + DESol({|----- _Y(x)| + x |---- _Y(x)| + _Y(x)}, {_Y(x)}) x + 1 | 3 | \ dx / \ dx / # 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)); 2 2 y(x) = _C1 exp(- 1/2 x ) + exp(- 1/2 x ) / / 2 \ | | d | / d \ 2 | 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)); ----- \ 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)); / ----- \ 1/2 1/2 | \ | y(x) = _C2 cos(2 x) + _C3 sin(2 x) + | ) _C1[_R] exp(_R x)| | / | | ----- | \_R = %1 / 3 %1 := RootOf(_Z + 2 _Z + 4) # 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)); / ----- \ / ----- \ | \ _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)); 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)); y(x) = _C1 exp(x) + _C2 exp(x) BesselJ(0, x) + _C3 exp(x) BesselY(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)); _C1 BesselJ(12, x) _C2 BesselY(12, x) y(x) = x + ------------------ + ------------------ x x # labahn15 dsolve(diff(y(x),x)+x*y(x)^2=1,y(x)); 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 ))