*----* MuPAD 1.3 -- Multi Processing Algebra Data Tool /| /| *----* | Copyright (c) 1992-96 by B. Fuchssteiner, Automath | *--|-* University of Paderborn. All rights reserved. |/ |/ *----* Demo version, please register with MuPAD-distribution@uni-paderborn.de # firstord1 # >> solve(ode((x^4-x^3)*diff(u(x),x) + 2*x^4*u(x) = x^3/3 + C,u(x))); { { C1 { ------------------------------------- { 2 { exp(2 x) - 2 x exp(2 x) + x exp(2 x) 2 exp(x) - -------------------------------------- + 2 2 2 2 4 (exp(x) - 2 x exp(x) + x exp(x) ) 2 x exp(x) -------------------------------------- + 2 2 2 2 6 (exp(x) - 2 x exp(x) + x exp(x) ) 2 } C exp(x) } ----------------------------------------- } 2 2 2 2 2 } 2 x (exp(x) - 2 x exp(x) + x exp(x) ) } # firstord2 # >> solve(ode(-1/2*diff(u(x),x)+u(x)=sin(x),u(x))); { 2 cos(x) 4 sin(x) } { -------- + -------- + C2 exp(2 x) } { 5 5 } # firstord3 # >> solve(ode(diff(y(x),x)=y(x)/(y(x)*ln(y(x))+x),y(x))); / 2 \ | x ln(y) | solve| C3 - - + ------, y | \ y 2 / # firstord4 # >> solve(ode(2*y(x)*diff(y(x),x)^2-2*x*diff(y(x),x)-y(x)=0,y(x))); { 1/2 2 1/2 1/2 2 1/2 { 6 (x ) 6 (x ) { ------------, - ------------, 0, RootOf( { 2 2 } 3 6 2 2 4 4 2 } - 2 x + 4 y(x) + 12 x y(x) - 12 x y(x) + 9 x y(x) - 2, y(x)) } } # bernoulli # >> solve(ode(diff(y(x),x)+y(x)=y(x)^3*sin(x),y(x))); { 1 } { ---------------------------------------- } { / 2 cos(x) 4 sin(x) \1/2 } { | -------- + -------- + C2 exp(2 x) | } { \ 5 5 / } # bernoulli2 # >> solve(ode(diff(y(x),x)+P(x)*y(x)=Q(x)*y(x)^(2/3),y(x))); { / { | { | / int(P(x), x) \ { | C3 exp| - ------------ | + { \ \ 3 / / int(P(x), x) \ / / int(P(x), x) \ \ \ } exp| - ------------ | int| Q(x) exp| ------------ |, x | | } \ 3 / \ \ 3 / / | } -------------------------------------------------------- |^(3) } 3 / } # clairaut # >> solve(ode((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 2 1/2 2 1/2 {- (- x + 1) , (- x + 1) , x C7 + (C7 + 1) , x C7 - (C7 + 1) } # clairaut2 # >> solve(ode(f(x*diff(y(x),x)-y(x))=g(diff(y(x),x)),y(x))); solve(f(- y + x C5) = -g(C5), y, 1) # exact1st # >> solve(ode(diff(y(x),x)=(3*x^2-y(x)^2-7)/(exp(y(x))+2*x*y(x)+1),y(x))); 3 2 solve(7 x + y + C9 + exp(y) - x + x y , y) # homogeneous # >> solve(ode(diff(y(x),x)=(2*x^3*y(x)-y(x)^4)/(x^4-2*x*y(x)^3),y(x))); 3 {x RootOf(x - C7 v5 + x v5 , v5)} # factor # >> solve(ode(diff(y(x),x)*(diff(y(x),x)+y(x))=x*(x+y(x)),y(x))); { 2 } { x } { - x + C9 exp(-x) + 1, C8 + -- } { 2 } # interchange # >> solve(ode(diff(y(x),x)=x/(x^2*y(x)^2+y(x)^5),y(x))); / / 3 \ / 3 \ / 3 \ \ | | 2 y | 2 | 2 y | 3 | 2 y | | | 3 exp| - ---- | x exp| - ---- | y exp| - ---- | | solve| \ 3 / \ 3 / \ 3 / | | C13 - --------------- - ---------------- - ----------------, y | \ 4 2 2 / # lagrange # >> solve(ode(y(x)=2*x*diff(y(x),x)-a*diff(y(x),x)^3,y(x))); 4 4 2 3 2 2 3 {RootOf(27 a y + 64 x C14 - 144 a x y C14 - 16 x y + 64 a C14 + 2 2 128 a x C14 , y)} # lagrange2 # >> solve(ode(y(x)=2*x*diff(y(x),x)-diff(y(x),x)^2,y(x))); 3 2 3 2 2 {RootOf(- 18 x y C12 + 4 y + 9 C12 + 12 x C12 - 3 x y , y)} # riccati # >> solve(ode(diff(y(x),x)=exp(x)*y(x)^2-y(x)+exp(-x),y(x))); { cos(x) - C16 sin(x) } { ---------------------------- } { exp(x) (sin(x) + C16 cos(x)) } # riccati2 # >> solve(ode(diff(y(x),x)=y(x)^2-x*y(x)+1,y(x))); { 1 } { x + ---------------------------------------------------------------, x } { / 2 \ / 2 \ } { | x | 1/2 | x | 1/2 } { C18 exp| - -- | + 1/2 I (2 PI) exp| - -- | erf(1/2 I x 2 ) } { \ 2 / \ 2 / } # separable # >> solve(ode(diff(y(x),x)=(9*x^8+1)/(y(x)^2+1),y(x))); 3 9 {RootOf(3 C1 + 3 x - 3 y - y + 3 x , y)} # solvablex # >> solve(ode(2*x*diff(y(x),x)+y(x)*diff(y(x),x)^2-y(x),y(x))); 2 1/2 2 1/2 {- (2 x C21 + C21 ) , (2 x C21 + C21 ) , I x, (- I) x} # solvabley # >> solve(ode(x-y(x)*diff(y(x),x)+x*diff(y(x),x)^2,y(x))); / / 2 \1/2 | | u4 | | u4 | --- - 4 | | 2 | 2 | / / 2 \1/2 \ solve| u4 \ x / | u4 | u4 | | | - ---- + ----------------- - ln| -- + | --- - 4 | | = C2 + ln(x), | 2 4 x | x | 2 | | \ 4 x \ \ x / / \ / / 2 \1/2 | | | u4 | | | u4 | --- - 4 | | | 2 | 2 | / / 2 \1/2 \ | union solve| u4 \ x / | u4 | u4 | | u4 | | - ---- - ----------------- + ln| -- + | --- - 4 | | | | 2 4 x | x | 2 | | / \ 4 x \ \ x / / \ | | | | = C3 + ln(x), u4 | | / # SecOrderChangevar # >> solve(ode((a*x+b)^2*diff(y(x),x,x)+4*a*(a*x+b)*diff(y(x),x)+2*a^2*y(x), y(x))); { C28 + x C27 } { - -------------------- } { 2 2 2 } { 2 a b x + b + a x } # adjoint # >> solve(ode((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))); { { C29 { ------------------------------------- { 2 { exp(2 x) - 2 x exp(2 x) + x exp(2 x) 2 exp(x) - -------------------------------------- + 2 2 2 2 4 (exp(x) - 2 x exp(x) + x exp(x) ) 2 x exp(x) -------------------------------------- 2 2 2 2 6 (exp(x) - 2 x exp(x) + x exp(x) ) 2 } C28 exp(x) } - ----------------------------------------- } 2 2 2 2 2 } 2 x (exp(x) - 2 x exp(x) + x exp(x) ) } # secondord1 # >> solve(ode((x^2-x)*diff(w(x),x,x)+(1-2*x^2)*diff(w(x),x)+(4*x-2)*w(x),w(x))); { 2 } { x C30 } { ------ + C31 exp(2 x) } { 2 } # autonomous # >> solve(ode(diff(y(x),x,x)-diff(y(x),x)=2*y(x)*diff(y(x),x),y(x))); { / 1/2 1/2 \ } { 1/2 | C2 (4 C1 - 1) x (4 C1 - 1) | } { (4 C1 - 1) tan| ---------------- + --------------- | } { \ 2 2 / } { ------------------------------------------------------- - 1/2, C3 } { 2 } # autonomous2 # >> solve(ode(diff(y(x),x,x)/y(x)-diff(y(x),x)^2/y(x)^2-1+1/y(x)^3=0,y(x))); / / 2 \ \ | | diff(y(x), x, x) 1 diff(y(x), x) | | solve| ode| ---------------- + ----- - -------------- - 1 = 0, y(x) | | | | y(x) 3 2 | | \ \ y(x) y(x) / / # ymissing # >> solve(ode(diff(y(x),x,x)+2*x*diff(y(x),x)=2*x,y(x))); { 1/2 } { PI C32 erf(x) } { x + C33 + ---------------- } { 2 } # diff # >> solve(ode(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))); { 2 2 } { C21 x C17 2 C18 3 } { ---, x C18 + ------ + ------, x C22 } { x 2 3 C17 } # equidimx # >> solve(ode(x*diff(y(x),x,x)=2*y(x)*diff(y(x),x),y(x))); { / 1/2 1/2 \ } { 1/2 | C4 (- 4 C3 - 1) ln(x) (- 4 C3 - 1) | } { (- 4 C3 - 1) tan| ------------------ + --------------------- | } { \ 2 2 / } { ----------------------------------------------------------------- - 1/2 } { 2 } # equidimy # >> solve(ode((1-x)*(y(x)*diff(y(x),x,x)-diff(y(x),x)^2)+x^2*y(x)^2,y(x))); { / 2 3 \ } { | x x | } { exp| C2 + -- + -- + x (C1 - 1) - ln(x - 1) + x ln(x - 1) | } { \ 2 6 / } # exact2nd # >> solve(ode(x*y(x)*diff(y(x),x,x)+x*diff(y(x),x)^2+y(x)*diff(y(x),x)=0,y(x))); 2 {RootOf(2 C43 + 2 C42 ln(x) - y , y)} # factoring # >> solve(ode(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))); {C44 exp(x) + C45 exp(-x), C46 exp(x) + x C47 exp(x)} # liouvillian # >> solve(ode((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))); { / 2 1/2 \ } { | C51 (- 2 x + x ) | } { | - ---------------------, x | } { | 2 3 | } { int| 2 x x | } { | - -------- + -------- | } { / 1 \ | / 1 \ / 1 \ | } { C52 exp| - - | | exp| - | exp| - | | } { \ x / \ \ x / \ x / / } { -------------- + --------------------------------- } { 1/2 / 1 \ 2 1/2 } { (x (x - 2)) exp| - | (- 2 x + x ) } { \ x / } # reduction # >> solve(ode(diff(y(x),x,x)-2*x*diff(y(x),x)+2*y(x)=3,y(x))); solve(ode(2 y(x) - 2 x diff(y(x), x) + diff(y(x), x, x) = 3, y(x))) # intfactors # >> solve(ode(sqrt(x)*diff(y(x),x,x)+2*x*diff(y(x),x)+3*y(x)=0,y(x))); 1/2 solve(ode(3 y(x) + 2 x diff(y(x), x) + x diff(y(x), x, x) = 0, y(x))) # scaleinv # >> solve(ode(x^2*diff(y(x),x,x)+3*x*diff(y(x),x)+2*y(x)-1/y(x)^3/x^4,y(x))); / / { 1 1 I - I } | | { -, - -, -, --- } union solve| int| { x x x x } \ \ x \ ----------------------------------------------------, u2 | 4 2 2 2 | = C2 + ln(x), RootOf((x u2) - 2 C1 (x u2) + (x u2) u3 + 1, u3) / \ | u2 | / # undet # >> solve(ode(diff(y(x),x,x)-2/x^2*y(x)=7*x^4+3*x^3,y(x))); { 5 6 } { C52 x x 2 } { - --- + -- + -- + x C53 } { 3 x 6 4 } # variation # >> solve(ode(diff(y(x),x,x)+y(x)=csc(x),y(x))); { 2 I x sin(x) { 2 I x sin(x) + C6 sin(x) + ----------------------------- + { cos(2 x) + (- I) sin(2 x) - 1 (- 2 I) C5 sin(x) ----------------------------- + sin(x) ln(cos(2 x) + (- I) sin(2 x) - 1) cos(2 x) + (- I) sin(2 x) - 1 } } } # constantcoeff # >> solve(ode(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))); {C59 exp(x) + x C60 exp(x) + C62 exp(4 x) + C56 exp(3 x) + C57 exp(2 x) + 2 x C58 exp(2 x) + x C61 exp(x)} # euler # >> solve(ode(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))); { C65 2 4 } { x C64 + --- + x C66 + x C67 } { x } # exactnth # >> solve(ode((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))); { 4 2 } { C70 x C71 x x C68 } { ---------- + ---------- + -------------- - -------------- } { 2 2 2 2 } { x + x + 1 x + x + 1 4 (x + x + 1) 2 (x + x + 1) } # circle # >> solve( ode( (diff(y(x),x)^2+1)*diff(y(x),x$3) - 3*diff(y(x),x)*diff(y(x),x$2)^2=0, y(x) ) ); 2 2 2 {C81, x C80 + C80 C83} union solve(int(1 / (RootOf(2 y C78 C79 + y C78 + 2 2 2 2 2 2 2 2 2 2 C78 C79 + 2 y C78 C79 u42 + y C78 u42 + C78 C79 u42 - 1, u42)) , y) = x + C82, y) # transfBernoulli # >> solve(ode(3*diff(y(x),x$2)*diff(y(x),x$4)-5*diff(y(x),x$3)^2=0,y(x))); { { / 3 \3/2 { C88 + x C87 + (x + C86) (- 4 x - 4 C86) | - ------------------- | , { \ 2 x C84 + 2 C84 C86 / 2 } x C85 } C88 + x C87 + ------ } 2 } # delay # >> solve(ode(diff(y(t),t)+a*y(t-1)=0,y(t))); Error: not an ordinary differential equation in the given variable [ode::solve_eq_irred] # several # >> solve(ode(diff(y(x,a),x)=a*y(x,a),y(x,a))); {C72 exp(a x)} # nthorder # >> solve(ode({diff(y(x),x$4)=sin(x),y(0)=0,D(y)(0)=0, D(D(y))(0)=0,D(D(D(y)))(0)=0},y(x))); { 3 } { x } { - x + sin(x) + -- } { 6 } # besselJ # >> solve(ode({x*diff(y(x),x,x)+diff(y(x),x)+2*x*y(x),y(0)=1,D(y)(0)=0},y(x))); solve(ode({2 x y(x) + diff(y(x), x) + x diff(y(x), x, x), D(y)(0) = 0, y(0) = 1}, y(x))) # separ # >> solve(ode({x*diff(y(x),x)^2-y(x)^2+1,y(0)=1},y(x))); 2 2 solve(ode({y(0) = 1, - y(x) + x diff(y(x), x) + 1}, y(x))) # ic2 # >> solve(ode({diff(y(x),x,x)+y(x)*diff(y(x),x)^3,y(0)=0,D(y)(0)=2},y(x))); 3 solve(ode({y(0) = 0, D(y)(0) = 2, diff(y(x), x, x) + y(x) diff(y(x), x) }, y(x))) # intcomb # >> solve(ode({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)})); 2 1/2 2 1/2 {[x(t) = - (- C5 + 3 z(t) ) , y(t) = - (- C4 - 3 z(t) ) ], 2 1/2 2 1/2 [x(t) = - (- C5 + 3 z(t) ) , y(t) = (- C4 - 3 z(t) ) ], 2 1/2 2 1/2 [x(t) = (- C5 + 3 z(t) ) , y(t) = - (- C4 - 3 z(t) ) ], 2 1/2 2 1/2 [x(t) = (- C5 + 3 z(t) ) , y(t) = (- C4 - 3 z(t) ) ]} # mriccati # >> solve(ode({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)})); 2 2 4 2 {x(t) = (- exp(c t) x(0) - exp(c t) x(0) int(a(t) exp(c t) , t) 2 2 2 2 - exp(c t) x(0) y(0) int(a(t) exp(c t) , t)) / ( 2 2 x(0) int(- a(t) exp(c t) , t) + y(0) int(b(t) exp(c t) , t) 3 2 2 2 - x(0) int(a(t) exp(c t) , t) + x(0) y(0) int(b(t) exp(c t) , t) 4 2 2 4 2 - x(0) int(b(t) exp(c t) , t) + x(0) int(a(t) exp(c t) , t) 2 2 2 2 2 int(- a(t) exp(c t) , t) - x(0) y(0) int(b(t) exp(c t) , t) + 2 2 2 2 x(0) y(0) int(a(t) exp(c t) , t) int(- a(t) exp(c t) , t) - 1) , y(t) 2 2 2 4 2 = (exp(c t) x(0) y(0) - exp(c t) x(0) int(b(t) exp(c t) , t) 2 2 2 2 - exp(c t) x(0) y(0) int(b(t) exp(c t) , t)) / ( 2 2 - x(0) int(- a(t) exp(c t) , t) - y(0) int(b(t) exp(c t) , t) + 3 2 2 2 x(0) int(a(t) exp(c t) , t) - x(0) y(0) int(b(t) exp(c t) , t) + 4 2 2 4 2 x(0) int(b(t) exp(c t) , t) - x(0) int(a(t) exp(c t) , t) 2 2 2 2 2 int(- a(t) exp(c t) , t) + x(0) y(0) int(b(t) exp(c t) , t) 2 2 2 2 - x(0) y(0) int(a(t) exp(c t) , t) int(- a(t) exp(c t) , t) + 1) } # vector # >> solve(ode({diff(x(t),t)=9*x(t)+2*y(t),diff(y(t),t)=x(t)+8*y(t)}, {x(t),y(t)})); {{y(t) = C35 exp(7 t) + C34 exp(10 t), x(t) = - C35 exp(7 t) + 2 C34 exp(10 t)}} # triangular # >> solve(ode({diff(x(t),t)=x(t)*(1+cos(t)/(2+sin(t))),diff(y(t),t)=x(t)-y(t)}, {x(t),y(t)})); { { C36 cos(t) exp(t) { { y(t) = C36 exp(t) + C37 exp(-t) - ----------------- + { { 5 2 C36 sin(t) exp(t) } } -------------------, x(t) = C36 exp(t) (sin(t) + 2) } } 5 } } # Higher order # >> solve( ode( { 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)} ) ); solve({- x(t) + 2 y(t) + diff(x(t), t) = 0, - 2 diff(y(t), t) + diff(x(t), t, t) = 2 t - cos(2 t)}, {x(t), y(t)}) # Inhomogeneous system # >> 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}: >> solve( ode( eq,{y1(x),y2(x)} ) ); / / { 2 2 | | { x y1(x) y2(x) (2 x + 1) solve| ode| { diff(y2(x), x) = - -------- + ---------------- + 1, | | { 2 2 \ \ { x + 1 x (x + 1) } \ \ 1 y1(x) y2(x) } | | diff(y1(x), x) = - - ---------- + ----------- }, {y1(x), y2(x)} | | x 2 2 2 } | | x (x + 1) x (x + 1) } / / # 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)=0: >> solve(ode(eq,y(x))); / / / 2 4 6 8 | | | 10952 x 5920 x 296 x 151 x 10 solve| ode| diff(y(x), x, x) | -------- + ------- + ------ + ------ + 2 x \ \ \ 9 9 3 3 \ / 10 12 | | 104 x 1/2 + x + 5476/9 | - y(x) | ------- + (- 148/3 I) 222 + / \ 25 8 1/2 6 1/2 x ((- 22/15 I) 222 + 274/25) + x ((- 68/15 I) 222 + 7754/75) + 4 1/2 2 1/2 x ((- 194/15 I) 222 + 11248/75) + x ((- 296/5 I) 222 + 29452/75) \ \ \ | | | - 10952/5 | = 0, y(x) | | / / /