REDUCE 3.6, 15-Jul-95 ... 1: load_package(odesolve)\$ 2: operator y; % firstord1 3: (x^4-x^3)*df(y(x),x,1) + 2*x^4*y(x) = x^3/3 + C; 3 4 3 4 3*c + x (x - x )*df(y(x),x,1) + 2*x *y(x)=---------- 3 4: odesolve(ws,y(x),x); 2 2*x 2*x 3 2*x 2 12*arbconst(1)*x + 6*e *c + 2*e *x - 3*e *x {y(x)=------------------------------------------------------} 2*x 4 2*x 3 2*x 2 12*e *x - 24*e *x + 12*e *x % firstord2 5: odesolve(-1/2*df(y(x),x)+y(x)=sin(x),y(x),x); 2*x 5*e *arbconst(2) + 2*cos(x) + 4*sin(x) {y(x)=------------------------------------------} 5 % firstord3 13: odesolve(df(y(x),x)=y(x)/(y(x)*log(y(x))+x),y(x),x); - df(y(x),x)*log(y(x))*y(x) - df(y(x),x)*x + y(x) {----------------------------------------------------=0} log(y(x))*y(x) + x % firstord4 14: odesolve(2*y(x)*df(y(x),x)^2-2*x*df(y(x),x)-y(x)=0,y(x),x); 2 { - 2*df(y(x),x) *y(x) + 2*df(y(x),x)*x + y(x)=0} % bernoulli 16: odesolve(df(y(x),x)+y(x)=y(x)^3*sin(x),y(x),x); 2*x 1 5*e *arbconst(3) + 2*cos(x) + 4*sin(x) {-------=------------------------------------------} 2 5 y(x) % bernoulli2: only for n integer 18: operator P\$ 19: operator Q\$ 20: odesolve(df(y(x),x)+P(x)*y(x)=Q(x)*y(x)^(2/3),y(x),x); 2/3 { - df(y(x),x) + y(x) *q(x) - p(x)*y(x)=0} 21: odesolve(df(y(x),x)+P(x)*y(x)=Q(x)*y(x)^n,y(x),x); n { - df(y(x),x) + y(x) *q(x) - p(x)*y(x)=0} 22: odesolve(df(y(x),x)+P(x)*y(x)=Q(x)*y(x),y(x),x); int(q(x),x) e *arbconst(4) {y(x)=--------------------------} int(p(x),x) e 23: odesolve(df(y(x),x)+P(x)*y(x)=Q(x)*y(x)^2,y(x),x); 1 int(p(x),x) int(p(x),x) q(x) {------=e *arbconst(5) - e *int(--------------,x)} y(x) int(p(x),x) 25: odesolve(df(y(x),x)+P(x)*y(x)=Q(x)*y(x)^7,y(x),x); 1 6*int(p(x),x) 6*int(p(x),x) q(x) {-------=e *arbconst(7) - 6*e *int(----------------,x)} 6 6*int(p(x),x) y(x) e 26: odesolve(df(y(x),x)+P(x)*y(x)=Q(x)*y(x)^(-7),y(x),x); 8*int(p(x),x) 8 arbconst(8) + 8*int(e *q(x),x) {y(x) =--------------------------------------------} 8*int(p(x),x) e % clairaut 27: odesolve((x^2-1)*df(y(x),x)^2-2*x*y(x)*df(y(x),x)+y(x)^2-1,y(x),x); 2 2 2 2 {df(y(x),x) *x - df(y(x),x) - 2*df(y(x),x)*y(x)*x + y(x) - 1=0} % clairaut2 28: operator f\$ 29: operator g\$ 30: odesolve(f(x*df(y(x),x)-y(x))=g(df(y(x),x)),y(x),x); { - f(df(y(x),x)*x - y(x)) + g(df(y(x),x))=0} % exact1st 31: odesolve(df(y(x),x)=(3*x^2-y(x)^2-7)/(exp(y(x))+2*x*y(x)+1),y(x),x); y(x) 2 2 - e *df(y(x),x) - 2*df(y(x),x)*y(x)*x - df(y(x),x) - y(x) + 3*x - 7 {---------------------------------------------------------------------------=0} y(x) e + 2*y(x)*x + 1 % but solved if numerator taken, and with "depend" 4: depend y,x; 5: odesolve((exp(y)+2*x*y+1)*df(y,x)=(3*x^2-y^2-7),y,x); y 3 2 {arbconst(1) - e + x - x*y - 7*x - y=0} % homogeneous 38: odesolve(df(y(x),x)=(2*x^3*y(x)-y(x)^4)/(x^4-2*x*y(x)^3),y(x),x); ***** y x invalid as integration variable % but works with operator notation: 2: depend y,x; 3: odesolve(df(y,x)=(2*x^3*y-y^4)/(x^4-2*x*y^3),y,x); 3 3 arbconst(1)*x*y + x + y {---------------------------=0} x*y % factor 39: odesolve(df(y(x),x)*(df(y(x),x)+y(x))=x*(x+y(x)),y(x),x); 2 2 { - df(y(x),x) - df(y(x),x)*y(x) + y(x)*x + x =0} % interchange 40: odesolve(df(y(x),x)=x/(x^2*y(x)^2+y(x)^5),y(x),x); 5 2 2 - df(y(x),x)*y(x) - df(y(x),x)*y(x) *x + x {-----------------------------------------------=0} 5 2 2 y(x) + y(x) *x % lagrange 41: odesolve(y(x)=2*x*df(y(x),x)-a*df(y(x),x)^3,y(x),x); 3 { - df(y(x),x) *a + 2*df(y(x),x)*x - y(x)=0} % lagrange2 42: odesolve(y(x)=2*x*df(y(x),x)-df(y(x),x)^2,y(x),x); 2 { - df(y(x),x) + 2*df(y(x),x)*x - y(x)=0} % riccati 43: odesolve(df(y(x),x)=exp(x)*y(x)^2-y(x)+exp(-x),y(x),x); x 2*x 2 x - e *df(y(x),x) + e *y(x) - e *y(x) + 1 {---------------------------------------------=0} x e % riccati2 44: odesolve(df(y(x),x)=y(x)^2-x*y(x)+1,y(x),x); 2 { - df(y(x),x) + y(x) - y(x)*x + 1=0} % separable 48: odesolve(df(y(x),x)=(9*x^8+1)/(y(x)^2+1),y(x),x); 2 8 - df(y(x),x)*y(x) - df(y(x),x) + 9*x + 1 {---------------------------------------------=0} 2 y(x) + 1 % but works as an operator 2: depend y,x; 5: odesolve(df(y,x)=(9*x^8+1)/(y^2+1),y,x); 9 3 3*arbconst(2) - 3*x - 3*x + y + 3*y {---------------------------------------=0} 3 % solvablex 49: odesolve(2*x*df(y(x),x)+y(x)*df(y(x),x)^2-y(x),y(x),x); 2 {df(y(x),x) *y(x) + 2*df(y(x),x)*x - y(x)=0} % solvabley 50: odesolve(x-y(x)*df(y(x),x)+x*df(y(x),x)^2,y(x),x); 2 {df(y(x),x) *x - df(y(x),x)*y(x) + x=0} % SecOrderChangevar 3: equ:=(a*x+b)^2*df(y(x),x,2)+4*a*(a*x+b)*df(y(x),x)+2*a^2*y(x)\$ 4: odesolve(equ=0,y(x),x); 2 2 2 { - df(y(x),x,2)*a *x - 2*df(y(x),x,2)*a*b*x - df(y(x),x,2)*b 2 2 - 4*df(y(x),x)*a *x - 4*df(y(x),x)*a*b - 2*y(x)*a =0} % adjoint 5: odesolve((x^2-x)*df(y(x),x,x)+(2*x^2+4*x-3)*df(y(x),x)+8*x*y(x)=1,y(x),x); 2 2 { - df(y(x),x,2)*x + df(y(x),x,2)*x - 2*df(y(x),x)*x - 4*df(y(x),x)*x + 3*df(y(x),x) - 8*y(x)*x + 1=0} % secondord1 6: odesolve((x^2-x)*df(y(x),x,x)+(1-2*x^2)*df(y(x),x)+(4*x-2)*y(x),y(x),x); 2 2 {df(y(x),x,2)*x - df(y(x),x,2)*x - 2*df(y(x),x)*x + df(y(x),x) + 4*y(x)*x - 2*y(x)=0} % autonomous 7: odesolve(df(y(x),x,x)-df(y(x),x)=2*y(x)*df(y(x),x),y(x),x); { - df(y(x),x,2) + 2*df(y(x),x)*y(x) + df(y(x),x)=0} % autonomous2 8: odesolve(df(y(x),x,x)/y(x)-df(y(x),x)^2/y(x)^2-1+1/y(x)^3=0,y(x),x); 2 2 3 - df(y(x),x,2)*y(x) + df(y(x),x) *y(x) + y(x) - 1 {------------------------------------------------------=0} 3 y(x) % ymissing 9: odesolve(df(y(x),x,x)+2*x*df(y(x),x)=2*x,y(x),x); { - df(y(x),x,2) - 2*df(y(x),x)*x + 2*x=0} % diff 10: equ:=2*y(x)*df(y(x),x,x)-df(y(x),x)^2=1/3*(df(y(x),x)-x*df(y(x),x,x))^2\$ 11: odesolve(equ,y(x),x); 2 2 {(df(y(x),x,2) *x - 2*df(y(x),x,2)*df(y(x),x)*x - 6*df(y(x),x,2)*y(x) 2 + 4*df(y(x),x) )/3=0} % equidimx 13: odesolve(x*df(y(x),x,x)=2*y(x)*df(y(x),x),y(x),x); { - df(y(x),x,2)*x + 2*df(y(x),x)*y(x)=0} % equidimy 14: odesolve((1-x)*(y(x)*df(y(x),x,x)-df(y(x),x)^2)+x^2*y(x)^2=0,y(x),x); 2 2 {df(y(x),x,2)*y(x)*x - df(y(x),x,2)*y(x) - df(y(x),x) *x + df(y(x),x) 2 2 - y(x) *x =0} % exact2nd 15: odesolve(x*y(x)*df(y(x),x,x)+x*df(y(x),x)^2+y(x)*df(y(x),x)=0,y(x),x); 2 { - df(y(x),x,2)*y(x)*x - df(y(x),x) *x - df(y(x),x)*y(x)=0} % factoring 16: equ:=df(y(x),x,2)^2-2*df(y(x),x)*df(y(x),x,2)+2*y(x)*df(y(x),x)-y(x)^2=0\$ 18: odesolve(equ,y(x),x); 2 2 { - df(y(x),x,2) + 2*df(y(x),x,2)*df(y(x),x) - 2*df(y(x),x)*y(x) + y(x) =0} % liouvillian 23: equ:=(x^3/2-x^2)*df(y(x),x,x)+(2*x^2-3*x+1)*df(y(x),x)+(x-1)*y(x)\$ 24: odesolve(equ,y(x),x); 3 2 2 {df(y(x),x,2)*x - 2*df(y(x),x,2)*x + 4*df(y(x),x)*x - 6*df(y(x),x)*x + 2*df(y(x),x) + 2*y(x)*x - 2*y(x)=0} % reduction 25: odesolve(df(y(x),x,x)-2*x*df(y(x),x)+2*y(x)=3,y(x),x); { - df(y(x),x,2) + 2*df(y(x),x)*x - 2*y(x) + 3=0} % intfactors 27: odesolve(sqrt(x)*df(y(x),x,x)+2*x*df(y(x),x)+3*y(x)=0,y(x),x); { - sqrt(x)*df(y(x),x,2) - 2*df(y(x),x)*x - 3*y(x)=0} % scaleinv 28: odesolve(x^2*df(y(x),x,x)+3*x*df(y(x),x)+2*y(x)-1/y(x)^3/x^4,y(x),x); 3 6 3 5 4 4 df(y(x),x,2)*y(x) *x + 3*df(y(x),x)*y(x) *x + 2*y(x) *x - 1 {----------------------------------------------------------------=0} 3 4 y(x) *x % undet 29: odesolve(df(y(x),x,x)-2/x^2*y(x)=7*x^4+3*x^3,y(x),x); 3 7 6 12*arbconst(3) + 12*arbconst(2)*x + 3*x + 2*x {y(x)=--------------------------------------------------} 12*x % variation: is the solution correct ? % in Maple: % y:=(-exp(I*x)*C5*sin(x)+exp(I*x)*C4*cos(x)-exp(I*x)-2*int(exp(2*I*x)* % int(1/(exp(I*x)*tan(x/2)^2-exp(I*x)),x),x))/exp(I*x): % combine(numer(convert(numer(diff(y,x,x)+y-csc(x)),exp)),exp); % combine(",exp); % 2 exp(4 I x) - 2 exp(2 I x) - 2 I exp(2 I x) - 2 I exp(4 I x) 31: odesolve(df(y(x),x,x)+y(x)=csc(x),y(x),x); i*x i*x i*x {y(x)=( - e *arbconst(5)*sin(x) + e *arbconst(4)*cos(x) - e 2*i*x 1 i*x - 2*int(e *int(-----------------------,x),x))/e } i*x x 2 i*x e *tan(---) - e 2 % constantcoeff 32: equ:=df(y(x),x,7)-14*df(y(x),x,6)+80*df(y(x),x,5)-242*df(y(x),x,4) +419*df(y(x),x,3)-416*df(y(x),x,2)+220*df(y(x),x)-48*y(x)=0\$ 33: odesolve(equ,y(x),x); 3*x 4*x 2*x {y(x)=e *arbconst(12) + e *arbconst(11) + e *arbconst(10)*x 2*x x 2 x x + e *arbconst(9) + e *arbconst(8)*x + e *arbconst(7)*x + e *arbconst(6)} % euler 34: odesolve(df(y(x),x,4)-4/x^2*df(y(x),x,x)+8/x^3*df(y(x),x)-8*y(x)/x^4,y(x),x); 2 3 5 arbconst(16) + arbconst(15)*x + arbconst(14)*x + arbconst(13)*x {y(x)=--------------------------------------------------------------------} x % exactnth 35: odesolve((1+x+x^2)*df(y(x),x,3)+(3+6*x)*df(y(x),x,x)+6*df(y(x),x)=6*x,y(x),x); 2 { - df(y(x),x,3)*x - df(y(x),x,3)*x - df(y(x),x,3) - 6*df(y(x),x,2)*x - 3*df(y(x),x,2) - 6*df(y(x),x) + 6*x=0} % circle 37: equ:=(df(y(x),x)^2+1)*df(y(x),x,3)-3*df(y(x),x)*df(y(x),x,x)^2\$ 38: odesolve(equ=0,y(x),x); 2 2 { - df(y(x),x,3)*df(y(x),x) - df(y(x),x,3) + 3*df(y(x),x,2) *df(y(x),x)=0} % transfBernoulli 41: odesolve(3*df(y(x),x,x)*df(y(x),x,4)-5*df(y(x),x,3)^2=0,y(x),x); 2 { - 3*df(y(x),x,4)*df(y(x),x,2) + 5*df(y(x),x,3) =0} % delay 42: odesolve(df(y(t),t)+a*y(t-1)=0,y(t),t); {y(t)=arbconst(17) - int(y(t - 1),t)*a} % several 43: odesolve(df(y(x,a),x)=a*y(x,a),y(x,a),x); a*x {y(x,a)=e *arbconst(18)} % 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; equ:=a2*df(y(x),x,x)-a0*y(x)=0; 7: odesolve(equ,y(x),x); 12 10 8 { - 225*df(y(x),x,2)*x - 450*df(y(x),x,2)*x - 11325*df(y(x),x,2)*x 6 4 2 - 22200*df(y(x),x,2)*x - 148000*df(y(x),x,2)*x - 273800*df(y(x),x,2)*x 8 6 - 136900*df(y(x),x,2) - 330*sqrt(222)*y(x)*i*x - 1020*sqrt(222)*y(x)*i*x 4 2 - 2910*sqrt(222)*y(x)*i*x - 13320*sqrt(222)*y(x)*i*x 10 8 6 - 11100*sqrt(222)*y(x)*i + 936*y(x)*x + 2466*y(x)*x + 23262*y(x)*x 4 2 + 33744*y(x)*x + 88356*y(x)*x - 492840*y(x)=0} % no command to solve boundary conditions ? % nthorder % besselJ % separ % ic2 % no command to solve systems ? % intcomb % moussiaux 3: odesolve(15*df(y(x),x)+24*y(x)^2=7*x^(-8/3),y(x),x); 2/3 2 2/3 2 2 - 15*x *df(y(x),x)*x - 24*x *y(x) *x + 7 {-------------------------------------------------=0} 2/3 2 x *x