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Codemist Standard Lisp 3.54 for DEC Alpha: May 23 1994 Dump file created: Mon May 23 10:39:11 1994 REDUCE 3.5, 15-Oct-93 ... Memory allocation: 6023424 bytes +++ About to read file tstlib.red % Initial load up list off echo$ .......................................................................... An example of the determination of point symmetries for ODEs -------------------------------------------------------------------------- This is LIEPDE - a program for calculating infinitesimal symmetries of single ODEs/PDEs and ODE/PDE - systems The ODE/PDE (-system) under investigation is : 4 3 2 df(y,x,2)*x - df(y,x)*x - 2*df(y,x)*x*y + 4*y 0 = -------------------------------------------------- 4 x for the function(s) : y(x) This is CRACK - a solver for overdetermined partial differential equations Version 25-08-1993 ************************************************************************** equations: expr. with 21 terms functions: eta_y(y,x) xi_x(y,x) separation w.r.t. y|1 new function: c1(x) new function: c2(x) integrated equation : 0=c1*y + c2 + xi_x separation yields 4 equations substitution : xi_x= - c1*y - c2 new function: c3(x) new function: c4(x) integrated equation : 3 2 2 2 3 3 3 0=3*df(c1,x)*x *y + 3*c1*x *y + 2*c1*y + 3*c3*x *y + 3*c4*x 3 + 3*eta_y*x substitution : eta_y 3 2 2 2 3 3 3 - 3*df(c1,x)*x *y - 3*c1*x *y - 2*c1*y - 3*c3*x *y - 3*c4*x =------------------------------------------------------------------ 3 3*x separation w.r.t. y separation yields 3 equations substitution : c1=0 substitution : - df(c2,x)*x + 3*c2 c3=---------------------- x substitution : 3 2 - 3*df(c2,x,2)*x + 5*df(c2,x)*x - 5*c2*x c4=--------------------------------------------- 2 separation w.r.t. y new constant: c5 new constant: c6 integrated equation : 0= - log(x)*c6*x + c2 - c5*x new constant: c7 new constant: c8 new constant: c9 integrated equation : 3 0= - log(x)*c8*x + c2 - c7*x - c9*x new constant: c10 new constant: c11 new constant: c12 new constant: c13 integrated equation : 2/3 2 2 2 0= - x *c12*x - log(x)*c11*x - c10*x - c13 + c2*x separation yields 3 equations substitution : c2=log(x)*c6*x + c5*x separation w.r.t. x linear independent expressions : x*log(x) x 3 x separation yields 3 equations substitution : c9=0 separation w.r.t. x linear independent expressions : 2 2/3 x *x 2 x *log(x) 2 x 1 separation yields 4 equations substitution : c12=0 substitution : c13=0 substitution : c10=c5 substitution : c11=c6 substitution : c8=c6 substitution : c7=c5 End of this CRACK run The solution : xi_x= - log(x)*c6*x - c5*x 2 eta_y= - 2*log(x)*c6*y - 2*c5*y - c6*x + c6*y Free functions or constants : c5 c6 ************************************************************************** CRACK needed : 6300 ms GC time : 500 ms The symmetries are: xi_x= - log(x)*c6*x - c5*x 2 eta_y= - 2*log(x)*c6*y - 2*c5*y - c6*x + c6*y with constants/functions: c6 c5 which are free. .......................................................................... An example of the determination of point symmetries for PDEs -------------------------------------------------------------------------- This is LIEPDE - a program for calculating infinitesimal symmetries of single ODEs/PDEs and ODE/PDE - systems The ODE/PDE (-system) under investigation is : 0 = df(u,x,2) - df(u,y) for the function(s) : u(y,x) This is CRACK - a solver for overdetermined partial differential equations Version 25-08-1993 ************************************************************************** equations: 0= - df(xi_y,u)*u|1 functions: eta_u(u,y,x) xi_y(u,y,x) xi_x(u,y,x) new function: c14(y,x) integrated equation : 0=c14 + xi_y substitution : xi_y= - c14 End of this CRACK run The solution : xi_y= - c14 Free functions or constants : xi_x(u,y,x) eta_u(u,y,x) c14(y,x) ************************************************************************** CRACK needed : 50 ms GC time : 234 ms This is CRACK - a solver for overdetermined partial differential equations Version 25-08-1993 ************************************************************************** equations: expr. with 14 terms functions: xi_x(u,y,x) eta_u(u,y,x) c14(y,x) separation w.r.t. u|1 separation w.r.t. u|2 separation w.r.t. u|1|2 separation w.r.t. u|2 new function: c15(y,x) new function: c16(y,x) integrated equation : 0=c15*u + c16 + xi_x new function: c17(y,x) integrated equation : 0=df(eta_u,u) - 2*df(xi_x,x) + c17 new function: c18(y,x) integrated equation : 0=c18 + xi_x new function: c19(y) integrated equation : 0=c14 + c19 separation yields 7 equations substitution : xi_x= - c18 substitution : c14= - c19 new function: c20(y) integrated equation : 0=df(c19,y)*x + 2*c18 + 2*c20 substitution : - df(c19,y)*x - 2*c20 c18=------------------------ 2 new function: c21(y,x) integrated equation : 0= - df(c19,y)*u + c17*u + c21 + eta_u substitution : eta_u=df(c19,y)*u - c17*u - c21 new function: c22(y) integrated equation : 2 0= - df(c19,y,2)*x - 4*df(c20,y)*x + 8*c17 + 8*c22 substitution : 2 df(c19,y,2)*x + 4*df(c20,y)*x - 8*c22 c17=---------------------------------------- 8 separation w.r.t. u separation yields 2 equations substitution : c15=0 separation w.r.t. u separation w.r.t. x new constant: c23 new constant: c24 new constant: c25 integrated equation : 2 0=c19 + c23*y + c24*y + c25 new constant: c26 new constant: c27 integrated equation : 0=c20 + c26*y + c27 new constant: c28 integrated equation : 0=5*df(c19,y) + 4*c22 + c28 separation yields 4 equations substitution : - 5*df(c19,y) - c28 c22=---------------------- 4 substitution : c20= - c26*y - c27 substitution : - df(c19,y)*x + 2*c26*y + 2*c27 c16=---------------------------------- 2 substitution : 2 c19= - c23*y - c24*y - c25 decoupling: c21 new equations: End of this CRACK run The solution : 2 c14=c23*y + c24*y + c25 2 - 4*c21 + c23*u*x + 2*c23*u*y + c24*u + 2*c26*u*x - c28*u eta_u=------------------------------------------------------------- 4 - 2*c23*x*y - c24*x - 2*c26*y - 2*c27 xi_x=---------------------------------------- 2 Remaining conditions : 0=df(c21,x,2) - df(c21,y) for the functions : c28 c23 c24 c25 c26 c27 c21(y,x) ************************************************************************** CRACK needed : 2000 ms GC time : 0 ms The symmetries are: - 2*c23*x*y - c24*x - 2*c26*y + 2*c27 xi_x=---------------------------------------- 2 2 xi_y= - c23*y - c24*y + c25 2 4*c21 + c23*u*x + 2*c23*u*y + c24*u + 2*c26*u*x + 4*c28 eta_u=---------------------------------------------------------- 4 with constants/functions: c21(y,x) c27 c26 c25 c24 c23 c28 which still have to satisfy: 0= - df(c21,x,2) + df(c21,y) .......................................................................... An example of the determination of first integrals of ODEs Determination of a first integral for: 2 2 2 df(y,x) *x - 2*df(y,x) - y df(y,x,2)=------------------------------ x new function: h_0(y,x) new function: h_1(y,x) new function: h_2(y,x) 2 of the type: df(y,x) *h_2 + df(y,x)*h_1 + h_0 This is CRACK - a solver for overdetermined partial differential equations Version 25-08-1993 ************************************************************************** equations: expr. with 13 terms functions: h_2(y,x) h_1(y,x) h_0(y,x) separation w.r.t. d_y(1) new function: c29(x) integrated equation : 2*x*y 0=e *h_2 - c29 separation yields 4 equations substitution : c29 h_2=-------- 2*x*y e substitution : df(h_0,x)*x h_1=------------- 2 y new function: c30(x) integrated equation : 2 2*x*y 3 x*y 3 2 0= - df(c29,x)*x*y + e *df(h_0,x)*x - e *c30*x *y 3 2 + 2*c29*x*y + 6*c29*y decoupling: h_0 new equations: expr. with 10 terms with leading derivative (df h_0 x 3) replaces a de with (df h_0 y) expr. with 20 terms with leading derivative (df h_0 x 2) replaces a de with (df h_0 x 3) expr. with 17 terms with leading derivative (df h_0 x) replaces a de with (df h_0 x 2) equations: expr. with 13 terms 2*x*y 2 2*x*y 2*x*y 2 0=e *df(h_0,x,2)*x - e *df(h_0,x)*x + e *df(h_0,y)*x*y 4 - 2*c29*y 2 2*x*y 3 x*y 3 2 0= - df(c29,x)*x*y + e *df(h_0,x)*x - e *c30*x *y 3 2 + 2*c29*x*y + 6*c29*y functions: c29 h_2=-------- 2*x*y e df(h_0,x)*x h_1=------------- 2 y h_0(y,x) c30(x) c29(x) separation w.r.t. y linear independent expressions : 1 y 2 y x*y e x*y y*e new constant: c31 integrated equation : 0=c30 - c31*x new constant: c32 new constant: c33 integrated equation : 0=c30 + c32*x + c33 new constant: c34 integrated equation : 4 0=c29 - c34*x new constant: c35 new constant: c36 integrated equation : 4 4 0= - log(x)*c36*x + c29 - c35*x new constant: c37 new constant: c38 new constant: c39 integrated equation : 6 4 3 0=c29 - c37*x - c38*x - c39*x separation yields 5 equations substitution : c30=c31*x separation w.r.t. x separation yields 2 equations substitution : c33=0 substitution : c32= - c31 substitution : 4 c29=c34*x separation w.r.t. x linear independent expressions : 4 x *log(x) 4 x separation yields 2 equations substitution : c36=0 new function: c40(y) integrated equation : 2*x*y 2*x*y x*y x*y 2 2 0=e *c40 + e *h_0 + e *c31*x*y + e *c31 - c34*x *y - 2*c34*x*y - c34 substitution : c35=c34 separation w.r.t. x separation yields 3 equations substitution : c39=0 substitution : c37=0 substitution : c38=c34 substitution : 2*x*y x*y x*y 2 2 h_0=( - e *c40 - e *c31*x*y - e *c31 + c34*x *y + 2*c34*x*y 2*x*y + c34)/e new constant: c41 integrated equation : 0=c40 + c41 substitution : c40= - c41 End of this CRACK run The solution : h_0 2*x*y x*y x*y 2 2 e *c41 - e *c31*x*y - e *c31 + c34*x *y + 2*c34*x*y + c34 =-------------------------------------------------------------------- 2*x*y e x*y 2 3 2 e *c31*x - 2*c34*x *y - 2*c34*x h_1=------------------------------------- 2*x*y e 4 c34*x h_2=-------- 2*x*y e Free functions or constants : c31 c34 c41 ************************************************************************** CRACK needed : 26899 ms GC time : 1650 ms 2 4 x*y 2 A first integral is: (df(y,x) *c34*x + e *df(y,x)*c31*x 3 2 x*y x*y - 2*df(y,x)*c34*x *y - 2*df(y,x)*c34*x - e *c31*x*y - e *c31 2 2 2*x*y + c34*x *y + 2*c34*x*y + c34)/e and an integrating factor: 2 2 x*y x *(2*df(y,x)*c34*x + e *c31 - 2*c34*x*y - 2*c34) ------------------------------------------------------ 2*x*y e free constants: c31 c34 .......................................................................... An example of the determination of a Lagrangian for an ODE Determination of a Lagrangian L for: 2 df(y,x,2)=x + 6*y 2 The ansatz: L = df(y,x) *u_ + v_ This is CRACK - a solver for overdetermined partial differential equations Version 25-08-1993 ************************************************************************** equations: 2 2 0=d_y(1) *df(u_,y) + 2*d_y(1)*df(u_,x) - df(v_,y) + 2*u_*x + 12*u_*y functions: u_(y,x) v_(y,x) separation w.r.t. d_y(1) new function: c42(y) integrated equation : 0=c42 + u_ new function: c43(x) integrated equation : 0=c43 + u_ separation yields 3 equations substitution : u_= - c43 generalized separation new constant: c44 substitution : c42= - c44 substitution : c43= - c44 new function: c45(x) integrated equation : 3 0= - 2*c44*x*y - 4*c44*y + c45 + v_ substitution : 3 v_=2*c44*x*y + 4*c44*y - c45 End of this CRACK run The solution : 3 v_=2*c44*x*y + 4*c44*y - c45 u_=c44 Free functions or constants : c45(x) c44 ************************************************************************** CRACK needed : 367 ms GC time : 0 ms 2 3 The solution: L = df(y,x) + 2*x*y + 4*y .......................................................................... An example of the factorization of an ODE Differential factorization of: 2 2 df(y,x) - df(y,x)*f*y - q*y df(y,x,2)=------------------------------- y The ansatz: df(y,x) = a#*y + b# This is CRACK - a solver for overdetermined partial differential equations Version 25-08-1993 ************************************************************************** equations: 2 2 2 0=( - df(a#,x)*y - df(b#,x)*y + a#*b#*y - a#*f*y + b# - b#*f*y 2 - q*y )/y functions: a#(x) b#(x) separation w.r.t. y new constant: c46 integrated equation : int(f,x) int(f,x) 0=e *a# + int(e *q,x) - c46 separation yields 3 equations substitution : b#=0 substitution : int(f,x) - int(e *q,x) + c46 a#=----------------------------- int(f,x) e End of this CRACK run The solution : b#=0 int(f,x) - int(e *q,x) + c46 a#=----------------------------- int(f,x) e Free functions or constants : c46 ************************************************************************** CRACK needed : 1683 ms GC time : 0 ms int(f,x) int(1/e ,x)*c46 e *c47 The solution y=-------------------------------------- int(f,x) int(f,x) int(int(e *q,x)/e ,x) e is the general solution of the original ODE (crack 67080 4084) End of Lisp run after 67.11+4.98 seconds