
Codemist Standard Lisp 3.54 for DEC Alpha: May 23 1994
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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
