# firstord1 # >> ode((x^4-x^3)*diff(u(x),x) + 2*x^4*u(x) = x^3/3 + C,u(x)): >> solve(%); { 1 x C C1 } { - ---------- + ---------- + ------------- + ---------------- } { 2 2 2 2 2 2 } { 4 (x - 1) 6 (x - 1) 2 x (x - 1) exp(x) (x - 1) } # firstord2 # >> ode(-1/2*diff(u(x),x)+u(x)=sin(x),u(x)): >> solve(%); { 2 cos(x) 4 sin(x) } { -------- + -------- + C2 exp(2 x) } { 5 5 } # firstord3 # >> ode(diff(y(x),x)=y(x)/(y(x)*ln(y(x))+x),y(x)): >> solve(%); / 2 \ | x ln(y) | solve| C3 - - + ------, y, 1, PrincipalValue | \ y 2 / # firstord4 # >> ode(2*y(x)*diff(y(x),x)^2-2*x*diff(y(x),x)-y(x)=0,y(x)): >> solve(%); 2 3 3 4 {0, RootOf(- y(x) + RootOf(12 x y(x) - 2 x + 4 y(x) + 9 x y(x) 2 2 2 2 - 12 x y(x) - 2, y(x)), y(x)), RootOf(3 x - 2 y(x) , y(x))} # bernoulli # >> ode(diff(y(x),x)+y(x)=y(x)^3*sin(x),y(x)): >> solve(%); { 1 } { ---------------------------------------- } { / 2 cos(x) 4 sin(x) \1/2 } { | -------- + -------- + C5 exp(2 x) | } { \ 5 5 / } # bernoulli2 # >> ode(diff(y(x),x)+P(x)*y(x)=Q(x)*y(x)^n,y(x)): >> solve(%); { / { | C1 exp(- int(P(x), x) + n int(P(x), x)) + { \ / Q(x) exp(int(P(x), x)) \ exp(- int(P(x), x) + n int(P(x), x)) int| ----------------------, x | - \ exp(n int(P(x), x)) / / Q(x) exp(int(P(x), x)) \ n exp(- int(P(x), x) + n int(P(x), x)) int| ----------------------, x | \ exp(n int(P(x), x)) / \ / 1 \ } |^| ------- | } / \ - n + 1 / } # clairaut # >> ode((x^2-1)*diff(y(x),x)^2-2*x*y(x)*diff(y(x),x)+y(x)^2-1,y(x)): >> solve(%); 2 2 2 2 2 2 {RootOf(x + y - 1, y), RootOf(- 2 C3 x y - C3 + y + C3 x - 1, y)} # clairaut2 # >> ode(f(x*diff(y(x),x)-y(x))=g(diff(y(x),x)),y(x)): >> solve(%); solve(f(- y + x C6) = g(C6), y, 1) # exact1st # >> ode(diff(y(x),x)=(3*x^2-y(x)^2-7)/(exp(y(x))+2*x*y(x)+1),y(x)): >> solve(%); 3 2 solve(C4 + 7 x + y + exp(y) - x + x y , y, 1, PrincipalValue) # homogeneous # >> ode(diff(y(x),x)=(2*x^3*y(x)-y(x)^4)/(x^4-2*x*y(x)^3),y(x)): >> solve(%); 3 {x RootOf(x - C7 v4 + x v4 , v4)} # factor # >> ode(diff(y(x),x)*(diff(y(x),x)+y(x))=x*(x+y(x)),y(x)): >> solve(%); { 2 } { x } { - x + C6 exp(-x) + 1, C5 + -- } { 2 } # interchange # >> ode(diff(y(x),x)=x/(x^2*y(x)^2+y(x)^5),y(x)): >> solve(%); / / 3 \ / 3 \ / 3 \ | | 2 y | 2 | 2 y | 3 | 2 y | | 3 exp| - ---- | x exp| - ---- | y exp| - ---- | solve| \ 3 / \ 3 / \ 3 / | C8 - --------------- - ---------------- - ----------------, y, 1, \ 4 2 2 \ | | | PrincipalValue | / # lagrange # >> ode(y(x)=2*x*diff(y(x),x)-a*diff(y(x),x)^3,y(x)): >> solve(%); 2 2 3 4 {RootOf(- y + RootOf(- 144 a x y C7 + 27 a y - 16 x y + 64 x C7 + 2 3 2 2 64 a C7 + 128 a x C7 , y), y)} # lagrange2 # >> ode(y(x)=2*x*diff(y(x),x)-diff(y(x),x)^2,y(x)): >> solve(%); 3 2 3 2 2 {RootOf(- 18 x y C9 + 4 y + 9 C9 + 12 x C9 - 3 x y , y)} # riccati # >> ode(diff(y(x),x)=exp(x)*y(x)^2-y(x)+exp(-x),y(x)): >> solve(%); { - cos(x) - C10 sin(x) } { ------------------------------ } { exp(x) (- sin(x) + C10 cos(x)) } # riccati2 # >> ode(diff(y(x),x)=y(x)^2-x*y(x)+1,y(x)): >> solve(%); { 1 } { x, x + ---------------------------------------------------------------- } { / 2 \ / 2 \ } { | x | 1/2 1/2 | x | 1/2 } { C12 exp| - -- | + 1/2 I PI 2 exp| - -- | erf(1/2 I x 2 ) } { \ 2 / \ 2 / } # separable # >> ode(diff(y(x),x)=(9*x^8+1)/(y(x)^2+1),y(x)): >> solve(%); 3 9 {RootOf(3 x - 3 y + 3 C13 - y + 3 x , y)} # solvablex # >> ode(2*x*diff(y(x),x)+y(x)*diff(y(x),x)^2-y(x),y(x)): >> solve(%); 2 1/2 2 1/2 {- (2 x C15 + C15 ) , (2 x C15 + C15 ) , I x, (- I) x} # solvabley # >> ode(x-y(x)*diff(y(x),x)+x*diff(y(x),x)^2,y(x)): >> solve(%); / / 1 \ | | -----------------------------, u14 | | | / / 2 \1/2 \ | | | | | u14 | | | solve| int| | | ---- - 4 | | | = C18 + ln(x), u14, 1, | | | | 2 | | | | | | u14 \ x / | | | | x | - --- - --------------- | | \ \ \ 2 x 2 / / \ / / 1 \ | | | -----------------------------, u14 | | | | / / 2 \1/2 \ | | | | | | u14 | | | PrincipalValue | union solve| int| | | ---- - 4 | | | | | | | | 2 | | | | | | | u14 \ x / | | | | | x | - --- + --------------- | | / \ \ \ 2 x 2 / / \ | | | = C19 + ln(x), u14, 1, PrincipalValue | | | | / # 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): >> ode(eq,y(x)); >> solve(%); { C21 + x C20 } { - -------------------- } { 2 2 2 } { 2 a b x + b + a x } # adjoint # >> 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)): >> solve(%); { 1 x C22 C23 } { - ---------- + ---------- - ------------- + ---------------- } { 2 2 2 2 2 2 } { 4 (x - 1) 6 (x - 1) 2 x (x - 1) exp(x) (x - 1) } # secondord1 # >> ode((x^2-x)*diff(w(x),x,x)+(1-2*x^2)*diff(w(x),x)+(4*x-2)*w(x),w(x)): >> solve(%); { 2 } { x C24 } { - ------ + C25 exp(2 x) } { 2 } # autonomous # >> ode(diff(y(x),x,x)-diff(y(x),x)=2*y(x)*diff(y(x),x),y(x)): >> solve(%); / / 1 2 C26 \ | | y + ------------------ - ---------------- + 1/2 | | ln| 1/2 1/2 | {C27} union solve| \ 2 (- 4 C26 + 1) (- 4 C26 + 1) / | - ----------------------------------------------------- | 1/2 \ (- 4 C26 + 1) / 1 2 C26 \ | y - ------------------ + ---------------- + 1/2 | ln| 1/2 1/2 | \ 2 (- 4 C26 + 1) (- 4 C26 + 1) / + ----------------------------------------------------- = x + C28, y, 1, 1/2 (- 4 C26 + 1) \ | | | PrincipalValue | | / # autonomous2 # >> ode(diff(y(x),x,x)/y(x)-diff(y(x),x)^2/y(x)^2-1+1/y(x)^3=0,y(x)): >> solve(%); / / 2 | | {1, RootOf(y + y + 1)} union solve| int| \ \ 1 \ -------------------------------------------------, y | 3 2 3 | = x + C30, y, 1, RootOf(6 y C29 - 3 y u17 + 6 y ln(y) + 2, u17) / \ | PrincipalValue | / # ymissing # >> ode(diff(y(x),x,x)+2*x*diff(y(x),x)=2*x,y(x)): >> solve(%); { 1/2 } { PI C31 erf(x) } { x + C32 + ---------------- } { 2 } # diff # >> 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)): >> solve(%); { 2 2 } { x C33 2 C34 3 C36 } { x C34 + ------ + ------, x C37, --- } { 2 3 C33 4 x } # equidimx # >> ode(x*diff(y(x),x,x)=2*y(x)*diff(y(x),x),y(x)): >> solve(%); / / 1 2 C38 \ | | y + ------------------ - ---------------- + 1/2 | | ln| 1/2 1/2 | solve| \ 2 (- 4 C38 + 1) (- 4 C38 + 1) / | - ----------------------------------------------------- + | 1/2 \ (- 4 C38 + 1) / 1 2 C38 \ | y - ------------------ + ---------------- + 1/2 | ln| 1/2 1/2 | \ 2 (- 4 C38 + 1) (- 4 C38 + 1) / ----------------------------------------------------- = C39 + ln(x), y, 1/2 (- 4 C38 + 1) \ | | | 1, PrincipalValue | | / # equidimy # >> ode((1-x)*(y(x)*diff(y(x),x,x)-diff(y(x),x)^2)+x^2*y(x)^2,y(x)): >> solve(%); {C41} # exact2nd # >> ode(x*y(x)*diff(y(x),x,x)+x*diff(y(x),x)^2+y(x)*diff(y(x),x)=0,y(x)): >> solve(%); 2 {RootOf(2 C44 + 2 C43 ln(x) - y , y)} # factoring # >> 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)): >> solve( % ); {C45 exp(x) + C46 exp(-x), C47 exp(x) + x C48 exp(x)} # liouvillian # >> 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)): >> solve(%); { / / 1 \ 2 1/2 \ } { | C49 exp| - | (- 2 x + x ) | } { | \ x / | } { int| - ----------------------------, x | } { | 2 3 | } { C50 \ - 2 x + x / } { ----------------------- + ---------------------------------------- } { / 1 \ 1/2 / 1 \ 2 1/2 } { exp| - | (x (x - 2)) exp| - | (- 2 x + x ) } { \ x / \ x / } # reduction # >> ode(diff(y(x),x,x)-2*x*diff(y(x),x)+2*y(x)=3,y(x)): >> solve(%); solve(ode(2 y(x) - 2 x diff(y(x), x) + diff(y(x), x, x) = 3, y(x))) # intfactors # >> ode(sqrt(x)*diff(y(x),x,x)+2*x*diff(y(x),x)+3*y(x)=0,y(x)): >> solve(%); Error: Illegal argument [pdivide] # scaleinv # >> 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)): >> solve(%); { 2 } / { 1 1 RootOf(u31 + 1) } | { -, - -, ---------------- } union solve| { x x x } \ / x \ | ----------------------------------------------------------, u31 | int| 4 2 2 2 | \ RootOf((x u31) - 2 C51 (x u31) + (x u31) u32 + 1, u32) / \ | = C52 + ln(x), u31, 1, PrincipalValue | / # undet # >> ode(diff(y(x),x,x)-2/x^2*y(x)=7*x^4+3*x^3,y(x)): >> solve(%); { 5 6 } { C53 x x 2 } { - --- + -- + -- + x C54 } { 3 x 6 4 } # variation # >> ode(diff(y(x),x,x)+y(x)=csc(x),y(x)): >> solve(%); { { 2 1/2 1/2 { C56 (- 4 cos(x) + 4) + (- 2 cos(2 x) + 2) { / x C55 \ } | ---------------------------- - ----------------------------, x | } int| 1/2 1/2 | } \ sin(x) (- 2 cos(2 x) + 2) sin(x) (- 2 cos(2 x) + 2) / } # constantcoeff # >> 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)): >> solve(%); {C60 exp(x) + x C61 exp(x) + C63 exp(4 x) + C57 exp(3 x) + C58 exp(2 x) + 2 x C59 exp(2 x) + x C62 exp(x)} # euler # >> 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)): >> solve(%); { C66 2 4 } { x C65 + --- + x C67 + x C68 } { x } # exactnth # >> 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)): >> solve(%); { 4 2 } { C72 x C70 x x C69 } { ---------- - ---------- + -------------- - -------------- } { 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) ) ); / | | {- C74 (- x - C78), C76} union solve| | \ / 1 \ \ | --------------------------, y | | | / 1 \1/2 | | int| | --------------- - 1 | | = x + C77, y, 1, PrincipalValue | | | 2 2 | | | \ \ C73 (y + C75) / / # transfBernoulli # >> solve(ode(3*diff(y(x),x$2)*diff(y(x),x$4)-5*diff(y(x),x$3)^2=0,y(x))); { 2 } { (x + C81) (- 4 x - 4 C81) x C80 } { C83 + x C82 + ----------------------------, C83 + x C82 + ------ } { / 2 x C79 2 C81 C79 \3/2 2 } { | - ------- - --------- | } { \ 3 3 / } # delay # >> ode(diff(y(t),t)+a*y(t-1)=0,y(t)): >> solve(%); 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))); {0} # nthorder # >> 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)): >> solve(%); { 3 } { x } { - x + sin(x) + -- } { 6 } # besselJ # >> 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(%); solve(ode({D(y)(0) = 0, y(0) = 1, 2 x y(x) + diff(y(x), x) + x diff(y(x), x, x)}, 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 # >> ode({diff(y(x),x,x)+y(x)*diff(y(x),x)^3,y(0)=0,D(y)(0)=2},y(x)): >> solve(%); 3 solve(ode({y(0) = 0, D(y)(0) = 2, diff(y(x), x, x) + y(x) diff(y(x), x) }, y(x))) # intcomb # >> 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)}): >> solve(%); 2 2 {[x(t) = RootOf(C93 - 3 z(t) + x(t) , x(t)), 2 2 y(t) = RootOf(C92 + 3 z(t) + y(t) , y(t))]} # mriccati # 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)}): >> solve(%); {x(t) = (exp(c t) x(0) - exp(c t) x(0) int(a(t) exp(c t) , t) 2 2 2 - exp(c t) y(0) int(a(t) exp(c t) , t)) / ( 2 2 - x(0) int(a(t) exp(c t) , t) - x(0) int(- a(t) exp(c t) , t) 2 2 2 2 2 2 - x(0) int(b(t) exp(c t) , t) - y(0) int(b(t) exp(c t) , t) + 2 2 2 x(0) int(a(t) exp(c t) , t) int(- a(t) exp(c t) , t) + 2 2 2 y(0) int(a(t) exp(c t) , t) int(- a(t) exp(c t) , t) + 1) , y(t) = ( 2 2 2 2 exp(c t) y(0) - exp(c t) x(0) int(b(t) exp(c t) , t) 2 2 2 - exp(c t) y(0) int(b(t) exp(c t) , t)) / ( 2 2 x(0) int(a(t) exp(c t) , t) + x(0) int(- a(t) exp(c t) , t) + 2 2 2 2 2 2 x(0) int(b(t) exp(c t) , t) + y(0) int(b(t) exp(c t) , t) 2 2 2 - x(0) int(a(t) exp(c t) , t) int(- a(t) exp(c t) , t) 2 2 2 - y(0) int(a(t) exp(c t) , t) int(- a(t) exp(c t) , t) - 1) } # vector # >> ode({diff(x(t),t)=9*x(t)+2*y(t),diff(y(t),t)=x(t)+8*y(t)},{x(t),y(t)}): >> solve(%); {{y(t) = C95 exp(7 t) + C94 exp(10 t), x(t) = - C95 exp(7 t) + 2 C94 exp(10 t)}} # triangular # >> 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)}): >> solve(%); { { C96 cos(t) exp(t) { { y(t) = C96 exp(t) + C97 exp(-t) - ----------------- + { { 5 2 C96 sin(t) exp(t) } } -------------------, x(t) = C96 exp(t) (sin(t) + 2) } } 5 } } # highOrder # >> 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(ode({- 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)} ) ); / / { | | { 1 y1(x) y2(x) solve| ode| { diff(y1(x), x) = - - ---------- + -----------, | | { x 2 2 2 \ \ { x (x + 1) x (x + 1) 2 2 } \ \ x y1(x) y2(x) (2 x + 1) } | | diff(y2(x), x) = - -------- + ---------------- + 1 }, {y1(x), y2(x)} | | 2 2 } | | 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))); Error: illegal coefficients [gcd] # end of file #