@@ -1,1408 +1,1408 @@
-REDUCE 3.6, 15-Jul-95, patched to 6 Mar 96 ...
-
-
-load_package crackapp;
-
-  % Needed for these tests.
-
-% 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
-0 = df(y,x,2)*x  - df(y,x)*x  - 2*df(y,x)*x*y + 4*y
-
-for the function(s) : 
-
-y(x)  
-
-
-time to formulate conditions: 190 ms    GC time : 0 ms
-
-This is CRACK - a solver for overdetermined partial differential equations
-Version 1995-03-20
-**************************************************************************
-
-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 + c2*y + xi_x
-
-
-separation yields 4 equations
-substitution :  
-xi_x= - c1 - c2*y
-
-eta_y(y,x)  
-xi_x= - c1 - c2*y
-
-
-new function: c3(x)  
-new function: c4(x)  
-new function: c5(x)  
-new function: c6(x)  
-integrated equation :  
-              3  2         2  2         3                            3
-0=3*df(c2,x)*x *y  + 3*c2*x *y  + 2*c2*y  + 3*c5 + 3*c6*y + 3*eta_y*x
-
-
-substitution :  
-                      3  2         2  2         3
-        - 3*df(c2,x)*x *y  - 3*c2*x *y  - 2*c2*y  - 3*c5 - 3*c6*y
-eta_y=------------------------------------------------------------
-                                     3
-                                  3*x
-
-
-                      3  2         2  2         3
-        - 3*df(c2,x)*x *y  - 3*c2*x *y  - 2*c2*y  - 3*c5 - 3*c6*y
-eta_y=------------------------------------------------------------
-                                     3
-                                  3*x
-
-
-xi_x= - c1 - c2*y
-
-
-separation w.r.t. y  
-separation yields 4 equations
-substitution :  
-c2=0
-
-
-        - c5 - c6*y
-eta_y=--------------
-             3
-            x
-
-
-xi_x= - c1
-
-
-substitution :  
-                3         2
-c6= - df(c1,x)*x  + 3*c1*x
-
-
-                 3           2
-       df(c1,x)*x *y - 3*c1*x *y - c5
-eta_y=--------------------------------
-                      3
-                     x
-
-
-xi_x= - c1
-
-
-substitution :  
-                     6               5         4
-     - 3*df(c1,x,2)*x  + 5*df(c1,x)*x  - 5*c1*x
-c5=----------------------------------------------
-                         2
-
-
-                     4               3                          2
-       3*df(c1,x,2)*x  - 5*df(c1,x)*x  + 2*df(c1,x)*x*y + 5*c1*x  - 6*c1*y
-eta_y=---------------------------------------------------------------------
-                                       2*x
-
-
-xi_x= - c1
-
-
-separation w.r.t. y  
-new constant: c7
-new constant: c8
-integrated equation :  
-0=log(x)*c8*x - c1 + c7*x
-
-
-new constant: c9
-new constant: c10
-new constant: c11
-integrated equation :  
-                           3
-0=log(x)*c10*x - c1 + c11*x  + c9*x
-
-
-new constant: c12
-new constant: c13
-new constant: c14
-new constant: c15
-integrated equation :  
-   2/3      2               2               2
-0=x   *c14*x  + log(x)*c13*x  - c1*x + c12*x  + c15
-
-
-separation yields 3 equations
-substitution :  
-c1=log(x)*c8*x + c7*x
-
-
-                                      2
-eta_y= - 2*log(x)*c8*y - 2*c7*y - c8*x  + c8*y
-
-
-xi_x= - log(x)*c8*x - c7*x
-
-
-separation w.r.t. x  
-linear independent expressions : 
-x*log(x)
-
-
- 3
-x
-
-
-x
-
-
-separation yields 3 equations
-substitution :  
-c11=0
-
-
-                                      2
-eta_y= - 2*log(x)*c8*y - 2*c7*y - c8*x  + c8*y
-
-
-xi_x= - log(x)*c8*x - c7*x
-
-
-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 :  
-c14=0
-
-
-                                      2
-eta_y= - 2*log(x)*c8*y - 2*c7*y - c8*x  + c8*y
-
-
-xi_x= - log(x)*c8*x - c7*x
-
-
-substitution :  
-c15=0
-
-
-                                      2
-eta_y= - 2*log(x)*c8*y - 2*c7*y - c8*x  + c8*y
-
-
-xi_x= - log(x)*c8*x - c7*x
-
-
-substitution :  
-c12=c7
-
-
-                                      2
-eta_y= - 2*log(x)*c8*y - 2*c7*y - c8*x  + c8*y
-
-
-xi_x= - log(x)*c8*x - c7*x
-
-
-substitution :  
-c13=c8
-
-
-                                      2
-eta_y= - 2*log(x)*c8*y - 2*c7*y - c8*x  + c8*y
-
-
-xi_x= - log(x)*c8*x - c7*x
-
-
-substitution :  
-c10=c8
-
-
-                                      2
-eta_y= - 2*log(x)*c8*y - 2*c7*y - c8*x  + c8*y
-
-
-xi_x= - log(x)*c8*x - c7*x
-
-
-substitution :  
-c9=c7
-
-
-                                      2
-eta_y= - 2*log(x)*c8*y - 2*c7*y - c8*x  + c8*y
-
-
-xi_x= - log(x)*c8*x - c7*x
-
-
-End of this CRACK run
-
-The solution : 
-xi_x= - log(x)*c8*x - c7*x
-
-
-                                      2
-eta_y= - 2*log(x)*c8*y - 2*c7*y - c8*x  + c8*y
-
-Free functions or constants : c7  c8  
-**************************************************************************
-
-
-CRACK needed :  2080 ms    GC time : 250 ms
-
-
-Remaining free functions after the last CRACK-run:
-c7  c8  
-
-
-The symmetries are:
-
-xi_x= - log(x)*c8*x - c7*x
-
-                                      2
-eta_y= - 2*log(x)*c8*y - 2*c7*y - c8*x  + c8*y
-
-with c7  c8  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)  
-
-
-time to formulate conditions: 170 ms    GC time : 0 ms
-
-This is CRACK - a solver for overdetermined partial differential equations
-Version 1995-03-20
-**************************************************************************
-
-equations: 
-0= - 2*df(xi_y,u)*u|1 - 2*df(xi_y,x)
-
-
-functions: eta_u(u,y,x)  xi_y(u,y,x)  xi_x(u,y,x)  
-separation w.r.t. u|1  
-new function: c18(u,y)  
-new function: c19(u,y)  
-integrated equation :  
-0=c19 + xi_y
-
-
-new function: c20(y,x)  
-new function: c21(y,x)  
-integrated equation :  
-0=c21 + xi_y
-
-
-separation yields 2 equations
-substitution :  
-xi_y= - c21
-
-eta_u(u,y,x)  
-xi_y= - c21
-
-xi_x(u,y,x)  
-generalized separation 
-new function: c22(y)  
-new function: c23(y)  
-separation yields 3 equations
-substitution :  
-c21=c22
-
-eta_u(u,y,x)  
-xi_y= - c22
-
-xi_x(u,y,x)  
-substitution :  
-c19=c22
-
-eta_u(u,y,x)  
-xi_y= - c22
-
-xi_x(u,y,x)  
-substitution :  
-c22= - c23
-
-eta_u(u,y,x)  
-xi_y=c23
-
-xi_x(u,y,x)  
-End of this CRACK run
-
-The solution : 
-xi_y=c23
-
-Free functions or constants : xi_x(u,y,x)  eta_u(u,y,x)  c23(y)  
-**************************************************************************
-
-
-CRACK needed :  120 ms    GC time : 0 ms
-
-
-Remaining free functions after the last CRACK-run:
-xi_x(u,y,x)  eta_u(u,y,x)  c23(y)  
-
-
-time to formulate conditions: 100 ms    GC time : 0 ms
-
-This is CRACK - a solver for overdetermined partial differential equations
-Version 1995-03-20
-**************************************************************************
-
-equations: expr. with 12 terms
-
-functions: xi_x(u,y,x)  eta_u(u,y,x)  c23(y)  
-separation w.r.t. u|1  
-separation w.r.t. u|2  
-separation w.r.t. u|2  
-new function: c24(y,x)  
-new function: c25(y,x)  
-new function: c26(y,x)  
-new function: c27(y,x)  
-integrated equation :  
-0=c26 + c27*u + xi_x
-
-
-new function: c28(y,x)  
-integrated equation :  
-0=df(eta_u,u) - 2*df(xi_x,x) + c28
-
-
-new function: c29(y,x)  
-new function: c30(y,x)  
-integrated equation :  
-0=c30 + xi_x
-
-
-new function: c31(u,y)  
-new function: c32(u,y)  
-integrated equation :  
-0=df(c23,y)*x + c32 - 2*xi_x
-
-
-separation yields 6 equations
-substitution :  
-xi_x= - c30
-
-
-xi_x= - c30
-
-eta_u(u,y,x)  c23(y)  
-generalized separation 
-new function: c33(y)  
-new function: c34(y)  
-separation yields 3 equations
-substitution :  
-c32= - c34
-
-
-xi_x= - c30
-
-eta_u(u,y,x)  c23(y)  
-substitution :  
-      - df(c23,y)*x + c34
-c30=----------------------
-              2
-
-
-      df(c23,y)*x - c34
-xi_x=-------------------
-              2
-
-eta_u(u,y,x)  c23(y)  
-substitution :  
-c33= - c34
-
-
-      df(c23,y)*x - c34
-xi_x=-------------------
-              2
-
-eta_u(u,y,x)  c23(y)  
-new function: c35(x,y)  
-new function: c36(x,y)  
-integrated equation :  
-0=df(c23,y)*u - c28*u + c36 - eta_u
-
-
-substitution :  
-eta_u=df(c23,y)*u - c28*u + c36
-
-
-      df(c23,y)*x - c34
-xi_x=-------------------
-              2
-
-
-eta_u=df(c23,y)*u - c28*u + c36
-
-c23(y)  
-new function: c37(y)  
-new function: c38(y)  
-integrated equation :  
-               2
-0=df(c23,y,2)*x  - 2*df(c34,y)*x - 8*c28 + 2*c38
-
-
-substitution :  
-                  2
-     df(c23,y,2)*x  - 2*df(c34,y)*x + 2*c38
-c28=----------------------------------------
-                       8
-
-
-      df(c23,y)*x - c34
-xi_x=-------------------
-              2
-
-
-                         2
-        - df(c23,y,2)*u*x  + 8*df(c23,y)*u + 2*df(c34,y)*u*x + 8*c36 - 2*c38*u
-eta_u=-------------------------------------------------------------------------
-                                          8
-
-c23(y)  
-separation w.r.t. u  
-separation yields 2 equations
-substitution :  
-c27=0
-
-
-      df(c23,y)*x - c34
-xi_x=-------------------
-              2
-
-
-                         2
-        - df(c23,y,2)*u*x  + 8*df(c23,y)*u + 2*df(c34,y)*u*x + 8*c36 - 2*c38*u
-eta_u=-------------------------------------------------------------------------
-                                          8
-
-c23(y)  
-separation w.r.t. u  
-separation w.r.t. x  
-new constant: c39
-new constant: c40
-new constant: c41
-new constant: c42
-new constant: c43
-new constant: c44
-integrated equation :  
-                       2
-0=2*c23 + 2*c42 + c43*y  + 2*c44*y
-
-
-new constant: c45
-new constant: c46
-new constant: c47
-new constant: c48
-integrated equation :  
-0=c34 + c47 + c48*y
-
-
-new constant: c49
-integrated equation :  
-0=5*df(c23,y) - c38 + c49
-
-
-separation yields 4 equations
-substitution :  
-c34= - c47 - c48*y
-
-
-      df(c23,y)*x + c47 + c48*y
-xi_x=---------------------------
-                  2
-
-
-                         2
-        - df(c23,y,2)*u*x  + 8*df(c23,y)*u + 8*c36 - 2*c38*u - 2*c48*u*x
-eta_u=-------------------------------------------------------------------
-                                       8
-
-c23(y)  
-substitution :  
-      - df(c23,y)*x - c47 - c48*y
-c26=------------------------------
-                  2
-
-
-      df(c23,y)*x + c47 + c48*y
-xi_x=---------------------------
-                  2
-
-
-                         2
-        - df(c23,y,2)*u*x  + 8*df(c23,y)*u + 8*c36 - 2*c38*u - 2*c48*u*x
-eta_u=-------------------------------------------------------------------
-                                       8
-
-c23(y)  
-substitution :  
-                     2
-      - 2*c42 - c43*y  - 2*c44*y
-c23=-----------------------------
-                  2
-
-
-       - c43*x*y - c44*x + c47 + c48*y
-xi_x=----------------------------------
-                     2
-
-
-                                2
-       8*c36 - 2*c38*u + c43*u*x  - 8*c43*u*y - 8*c44*u - 2*c48*u*x
-eta_u=--------------------------------------------------------------
-                                    8
-
-
-                     2
-      - 2*c42 - c43*y  - 2*c44*y
-c23=-----------------------------
-                  2
-
-
-substitution :  
-c38= - 5*c43*y - 5*c44 + c49
-
-
-       - c43*x*y - c44*x + c47 + c48*y
-xi_x=----------------------------------
-                     2
-
-
-                      2
-       8*c36 + c43*u*x  + 2*c43*u*y + 2*c44*u - 2*c48*u*x - 2*c49*u
-eta_u=--------------------------------------------------------------
-                                    8
-
-
-                     2
-      - 2*c42 - c43*y  - 2*c44*y
-c23=-----------------------------
-                  2
-
-
-decoupling: 
-c36
-
-
-new equations: 
-End of this CRACK run
-
-The solution : 
-                     2
-      - 2*c42 - c43*y  - 2*c44*y
-c23=-----------------------------
-                  2
-
-
-                      2
-       8*c36 + c43*u*x  + 2*c43*u*y + 2*c44*u - 2*c48*u*x - 2*c49*u
-eta_u=--------------------------------------------------------------
-                                    8
-
-
-       - c43*x*y - c44*x + c47 + c48*y
-xi_x=----------------------------------
-                     2
-
-Remaining conditions : 
-0=df(c36,x,2) - df(c36,y)
-
-for the functions : c42  c47  c48  c43  c44  
-c49  c36(x,y)  
-**************************************************************************
-
-
-CRACK needed :  980 ms    GC time : 110 ms
-
-
-Remaining free functions after the last CRACK-run:
-c42  c47  c48  c43  c44  
-c49  c36(x,y)  
-
-
-Free constants and/or functions have been rescaled.
-
-The symmetries are:
-
-xi_x= - 4*c43*x*y - 2*c44*x + c47 + 2*c48*y
-
-                     2
-xi_y= - c42 - 4*c43*y  - 4*c44*y
-
-                   2
-eta_u=c36 + c43*u*x  + 2*c43*u*y + c44*u - c48*u*x - c49*u
-
-with c42  c47  c48  c43  c44  
-c49  c36(x,y)  
-which still have to satisfy: 
-
-0=df(c36,x,2) - df(c36,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 1995-03-20
-**************************************************************************
-
-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: c50(x)  
-integrated equation :  
-   2*x*y
-0=e     *h_2 - c50
-
-
-separation yields 4 equations
-substitution :  
-      c50
-h_2=--------
-      2*x*y
-     e
-
-
-      c50
-h_2=--------
-      2*x*y
-     e
-
-h_1(y,x)  h_0(y,x)  
-substitution :  
-     df(h_0,x)*x
-h_1=-------------
-          2
-         y
-
-
-      c50
-h_2=--------
-      2*x*y
-     e
-
-
-     df(h_0,x)*x
-h_1=-------------
-          2
-         y
-
-h_0(y,x)  
-new function: c51(x)  
-integrated equation :  
-               2    2*x*y            3    x*y      3  2            3          2
-0=df(c50,x)*x*y  - e     *df(h_0,x)*x  + e   *c51*x *y  - 2*c50*x*y  - 6*c50*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          4
-0=e     *df(h_0,x,2)*x  - e     *df(h_0,x)*x + e     *df(h_0,y)*x*y  - 2*c50*y
-
-
-               2    2*x*y            3    x*y      3  2            3          2
-0=df(c50,x)*x*y  - e     *df(h_0,x)*x  + e   *c51*x *y  - 2*c50*x*y  - 6*c50*y
-
-
-functions: 
-      c50
-h_2=--------
-      2*x*y
-     e
-
-
-     df(h_0,x)*x
-h_1=-------------
-          2
-         y
-
-h_0(y,x)  c51(x)  c50(x)  
-separation w.r.t. y  
-linear independent expressions : 
-1
-
-
-y
-
-
- 2
-y
-
-
- x*y
-e
-
-
-   x*y
-y*e
-
-
-new constant: c52
-integrated equation :  
-0=c51 - c52*x
-
-
-new constant: c53
-new constant: c54
-integrated equation :  
-0=c51 + c53 + c54*x
-
-
-new constant: c55
-integrated equation :  
-             4
-0=c50 - c55*x
-
-
-new constant: c56
-new constant: c57
-integrated equation :  
-              4              4
-0=log(x)*c57*x  - c50 + c56*x
-
-
-new constant: c58
-new constant: c59
-new constant: c60
-integrated equation :  
-             6        4        3
-0=c50 - c58*x  - c59*x  - c60*x
-
-
-separation yields 5 equations
-substitution :  
-c51=c52*x
-
-
-      c50
-h_2=--------
-      2*x*y
-     e
-
-
-     df(h_0,x)*x
-h_1=-------------
-          2
-         y
-
-h_0(y,x)  
-separation w.r.t. x  
-separation yields 2 equations
-substitution :  
-c53=0
-
-
-      c50
-h_2=--------
-      2*x*y
-     e
-
-
-     df(h_0,x)*x
-h_1=-------------
-          2
-         y
-
-h_0(y,x)  
-substitution :  
-         4
-c50=c55*x
-
-
-          4
-     c55*x
-h_2=--------
-      2*x*y
-     e
-
-
-     df(h_0,x)*x
-h_1=-------------
-          2
-         y
-
-h_0(y,x)  
-separation w.r.t. x  
-linear independent expressions : 
- 4
-x *log(x)
-
-
- 4
-x
-
-
-separation yields 2 equations
-substitution :  
-c57=0
-
-
-          4
-     c55*x
-h_2=--------
-      2*x*y
-     e
-
-
-     df(h_0,x)*x
-h_1=-------------
-          2
-         y
-
-h_0(y,x)  
-new function: c61(y)  
-substitution :  
-c56=c55
-
-
-          4
-     c55*x
-h_2=--------
-      2*x*y
-     e
-
-
-     df(h_0,x)*x
-h_1=-------------
-          2
-         y
-
-h_0(y,x)  
-substitution :  
-c54= - c52
-
-
-          4
-     c55*x
-h_2=--------
-      2*x*y
-     e
-
-
-     df(h_0,x)*x
-h_1=-------------
-          2
-         y
-
-h_0(y,x)  
-separation w.r.t. x  
-separation yields 3 equations
-substitution :  
-c60=0
-
-
-          4
-     c55*x
-h_2=--------
-      2*x*y
-     e
-
-
-     df(h_0,x)*x
-h_1=-------------
-          2
-         y
-
-h_0(y,x)  
-substitution :  
-c58=0
-
-
-          4
-     c55*x
-h_2=--------
-      2*x*y
-     e
-
-
-     df(h_0,x)*x
-h_1=-------------
-          2
-         y
-
-h_0(y,x)  
-substitution :  
-c59=c55
-
-
-          4
-     c55*x
-h_2=--------
-      2*x*y
-     e
-
-
-     df(h_0,x)*x
-h_1=-------------
-          2
-         y
-
-h_0(y,x)  
-decoupling: 
-h_0
-
-
-new equations: 
-new function: c62(y)  
-new function: c63(y)  
-new function: c64(y)  
-new function: c65(y)  
-integrated equation :  expr. with 10 terms
-
-substitution :  
-      2*sqrt(2)*x*y + 2*x*y                  sqrt(2)*x*y + 2*x*y
-h_0=(e                     *sqrt(2)*c62 - 2*e                   *c65*y
-
-           sqrt(2)*x*y + x*y        2      sqrt(2)*x*y + x*y
-      - 2*e                 *c52*x*y  - 2*e                 *c52*y
-
-           sqrt(2)*x*y      2  3      sqrt(2)*x*y        2
-      + 2*e           *c55*x *y  + 4*e           *c55*x*y
-
-           sqrt(2)*x*y          2*x*y                  sqrt(2)*x*y + 2*x*y
-      + 2*e           *c55*y - e     *sqrt(2)*c63)/(2*e                   *y)
-
-
-          4
-     c55*x
-h_2=--------
-      2*x*y
-     e
-
-
-      2*sqrt(2)*x*y + 2*x*y          sqrt(2)*x*y + x*y      2  2
-h_1=(e                     *c62*x + e                 *c52*x *y
-
-           sqrt(2)*x*y      3  3      sqrt(2)*x*y      2  2    2*x*y
-      - 2*e           *c55*x *y  - 2*e           *c55*x *y  + e     *c63*x)/(
-
-    sqrt(2)*x*y + 2*x*y  2
-   e                   *y )
-
-
-h_0= expr. with 9 terms
-
-equations: 
-    2*sqrt(2)*x*y + 2*x*y        2*x*y
-   e                     *c62 + e     *c63
-0=-----------------------------------------
-                 sqrt(2)*x*y
-                e
-
-expr. with 12 terms
-
-functions: 
-          4
-     c55*x
-h_2=--------
-      2*x*y
-     e
-
-
-      2*sqrt(2)*x*y + 2*x*y          sqrt(2)*x*y + x*y      2  2
-h_1=(e                     *c62*x + e                 *c52*x *y
-
-           sqrt(2)*x*y      3  3      sqrt(2)*x*y      2  2    2*x*y
-      - 2*e           *c55*x *y  - 2*e           *c55*x *y  + e     *c63*x)/(
-
-    sqrt(2)*x*y + 2*x*y  2
-   e                   *y )
-
-
-h_0= expr. with 9 terms
-c62(y)  c63(y)  c65(y)  
-separation w.r.t. x  
-linear independent expressions : 
- 2*x*y + 2*sqrt(2)*x*y
-e
-
-
- 2*x*y
-e
-
-
-separation yields 2 equations
-substitution :  
-c62=0
-
-
-          4
-     c55*x
-h_2=--------
-      2*x*y
-     e
-
-
-      sqrt(2)*x*y + x*y      2  2      sqrt(2)*x*y      3  3
-h_1=(e                 *c52*x *y  - 2*e           *c55*x *y
-
-           sqrt(2)*x*y      2  2    2*x*y          sqrt(2)*x*y + 2*x*y  2
-      - 2*e           *c55*x *y  + e     *c63*x)/(e                   *y )
-
-
-           sqrt(2)*x*y + 2*x*y            sqrt(2)*x*y + x*y        2
-h_0=( - 2*e                   *c65*y - 2*e                 *c52*x*y
-
-           sqrt(2)*x*y + x*y            sqrt(2)*x*y      2  3
-      - 2*e                 *c52*y + 2*e           *c55*x *y
-
-           sqrt(2)*x*y        2      sqrt(2)*x*y          2*x*y
-      + 4*e           *c55*x*y  + 2*e           *c55*y - e     *sqrt(2)*c63)/(2
-
-     sqrt(2)*x*y + 2*x*y
-   *e                   *y)
-
-
-substitution :  
-c63=0
-
-
-          4
-     c55*x
-h_2=--------
-      2*x*y
-     e
-
-
-      x*y      2          3            2
-     e   *c52*x  - 2*c55*x *y - 2*c55*x
-h_1=-------------------------------------
-                    2*x*y
-                   e
-
-
-         2*x*y        x*y            x*y            2  2
-      - e     *c65 - e   *c52*x*y - e   *c52 + c55*x *y  + 2*c55*x*y + c55
-h_0=-----------------------------------------------------------------------
-                                     2*x*y
-                                    e
-
-
-new constant: c66
-integrated equation :  
-0=c65 + c66
-
-
-substitution :  
-c65= - c66
-
-
-          4
-     c55*x
-h_2=--------
-      2*x*y
-     e
-
-
-      x*y      2          3            2
-     e   *c52*x  - 2*c55*x *y - 2*c55*x
-h_1=-------------------------------------
-                    2*x*y
-                   e
-
-
-      2*x*y        x*y            x*y            2  2
-     e     *c66 - e   *c52*x*y - e   *c52 + c55*x *y  + 2*c55*x*y + c55
-h_0=--------------------------------------------------------------------
-                                    2*x*y
-                                   e
-
-
-End of this CRACK run
-
-The solution : 
-      2*x*y        x*y            x*y            2  2
-     e     *c66 - e   *c52*x*y - e   *c52 + c55*x *y  + 2*c55*x*y + c55
-h_0=--------------------------------------------------------------------
-                                    2*x*y
-                                   e
-
-
-      x*y      2          3            2
-     e   *c52*x  - 2*c55*x *y - 2*c55*x
-h_1=-------------------------------------
-                    2*x*y
-                   e
-
-
-          4
-     c55*x
-h_2=--------
-      2*x*y
-     e
-
-Free functions or constants : c52  c55  c66  
-**************************************************************************
-
-
-CRACK needed :  8530 ms    GC time : 570 ms
-                              2      4    x*y              2
-A first integral is:  (df(y,x) *c55*x  + e   *df(y,x)*c52*x
-
-                     3                    2    x*y            x*y
-    - 2*df(y,x)*c55*x *y - 2*df(y,x)*c55*x  - e   *c52*x*y - e   *c52
-
-           2  2                     2*x*y
-    + c55*x *y  + 2*c55*x*y + c55)/e
-
-and an integrating factor:  
-
-  2                 2    x*y
- x *(2*df(y,x)*c55*x  + e   *c52 - 2*c55*x*y - 2*c55)
-------------------------------------------------------
-                         2*x*y
-                        e
-
-free constants: c52  c55  
-..........................................................................
-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 1995-03-20
-**************************************************************************
-
-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: c67(y)  
-new function: c68(y)  
-integrated equation :  
-0=c68 + u_
-
-
-new function: c69(x)  
-new function: c70(x)  
-integrated equation :  
-0=c70 + u_
-
-
-separation yields 3 equations
-substitution :  
-u_= - c70
-
-
-u_= - c70
-
-v_(y,x)  
-generalized separation 
-new constant: c71
-new constant: c72
-separation yields 3 equations
-substitution :  
-c68= - c71
-
-
-u_= - c70
-
-v_(y,x)  
-substitution :  
-c70= - c71
-
-
-u_=c71
-
-v_(y,x)  
-substitution :  
-c71= - c72
-
-
-u_= - c72
-
-v_(y,x)  
-new function: c73(x)  
-new function: c74(x)  
-integrated equation :  
-                     3
-0=2*c72*x*y + 4*c72*y  + c74 + v_
-
-
-substitution :  
-                         3
-v_= - 2*c72*x*y - 4*c72*y  - c74
-
-
-u_= - c72
-
-
-                         3
-v_= - 2*c72*x*y - 4*c72*y  - c74
-
-
-End of this CRACK run
-
-The solution : 
-                         3
-v_= - 2*c72*x*y - 4*c72*y  - c74
-
-
-u_= - c72
-
-Free functions or constants : c74(x)  c72  
-**************************************************************************
-
-
-CRACK needed :  240 ms    GC time : 0 ms
-                                 2             2
-The solution:  L =  - (2*(x + 2*y )*y + df(y,x) )
-..........................................................................
-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 1995-03-20
-**************************************************************************
-
-equations: 
-            2                                2     2               2
-0=df(a#,x)*y  + df(b#,x)*y - a#*b#*y + a#*f*y  - b#  + b#*f*y + q*y
-
-
-functions: a#(x)  b#(x)  
-separation w.r.t. y  
-new constant: c75
-integrated equation :  
-   int(f,x)           int(f,x)
-0=e        *a# + int(e        *q,x) - c75
-
-
-separation yields 3 equations
-substitution :  
-b#=0
-
-a#(x)  
-b#=0
-
-
-substitution :  
-            int(f,x)
-     - int(e        *q,x) + c75
-a#=-----------------------------
-              int(f,x)
-             e
-
-
-            int(f,x)
-     - int(e        *q,x) + c75
-a#=-----------------------------
-              int(f,x)
-             e
-
-
-b#=0
-
-
-End of this CRACK run
-
-The solution : 
-b#=0
-
-
-            int(f,x)
-     - int(e        *q,x) + c75
-a#=-----------------------------
-              int(f,x)
-             e
-
-Free functions or constants : c75  
-**************************************************************************
-
-
-CRACK needed :  530 ms    GC time : 0 ms
-                            int(f,x)
-                     int(1/e        ,x)*c75
-                    e                      *c76
-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
-(TIME:  crack 23309 24969)
+REDUCE 3.6, 15-Jul-95, patched to 6 Mar 96 ...
+
+
+load_package crackapp;
+
+  % Needed for these tests.
+
+% 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
+0 = df(y,x,2)*x  - df(y,x)*x  - 2*df(y,x)*x*y + 4*y
+
+for the function(s) : 
+
+y(x)  
+
+
+time to formulate conditions: 190 ms    GC time : 0 ms
+
+This is CRACK - a solver for overdetermined partial differential equations
+Version 1995-03-20
+**************************************************************************
+
+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 + c2*y + xi_x
+
+
+separation yields 4 equations
+substitution :  
+xi_x= - c1 - c2*y
+
+eta_y(y,x)  
+xi_x= - c1 - c2*y
+
+
+new function: c3(x)  
+new function: c4(x)  
+new function: c5(x)  
+new function: c6(x)  
+integrated equation :  
+              3  2         2  2         3                            3
+0=3*df(c2,x)*x *y  + 3*c2*x *y  + 2*c2*y  + 3*c5 + 3*c6*y + 3*eta_y*x
+
+
+substitution :  
+                      3  2         2  2         3
+        - 3*df(c2,x)*x *y  - 3*c2*x *y  - 2*c2*y  - 3*c5 - 3*c6*y
+eta_y=------------------------------------------------------------
+                                     3
+                                  3*x
+
+
+                      3  2         2  2         3
+        - 3*df(c2,x)*x *y  - 3*c2*x *y  - 2*c2*y  - 3*c5 - 3*c6*y
+eta_y=------------------------------------------------------------
+                                     3
+                                  3*x
+
+
+xi_x= - c1 - c2*y
+
+
+separation w.r.t. y  
+separation yields 4 equations
+substitution :  
+c2=0
+
+
+        - c5 - c6*y
+eta_y=--------------
+             3
+            x
+
+
+xi_x= - c1
+
+
+substitution :  
+                3         2
+c6= - df(c1,x)*x  + 3*c1*x
+
+
+                 3           2
+       df(c1,x)*x *y - 3*c1*x *y - c5
+eta_y=--------------------------------
+                      3
+                     x
+
+
+xi_x= - c1
+
+
+substitution :  
+                     6               5         4
+     - 3*df(c1,x,2)*x  + 5*df(c1,x)*x  - 5*c1*x
+c5=----------------------------------------------
+                         2
+
+
+                     4               3                          2
+       3*df(c1,x,2)*x  - 5*df(c1,x)*x  + 2*df(c1,x)*x*y + 5*c1*x  - 6*c1*y
+eta_y=---------------------------------------------------------------------
+                                       2*x
+
+
+xi_x= - c1
+
+
+separation w.r.t. y  
+new constant: c7
+new constant: c8
+integrated equation :  
+0=log(x)*c8*x - c1 + c7*x
+
+
+new constant: c9
+new constant: c10
+new constant: c11
+integrated equation :  
+                           3
+0=log(x)*c10*x - c1 + c11*x  + c9*x
+
+
+new constant: c12
+new constant: c13
+new constant: c14
+new constant: c15
+integrated equation :  
+   2/3      2               2               2
+0=x   *c14*x  + log(x)*c13*x  - c1*x + c12*x  + c15
+
+
+separation yields 3 equations
+substitution :  
+c1=log(x)*c8*x + c7*x
+
+
+                                      2
+eta_y= - 2*log(x)*c8*y - 2*c7*y - c8*x  + c8*y
+
+
+xi_x= - log(x)*c8*x - c7*x
+
+
+separation w.r.t. x  
+linear independent expressions : 
+x*log(x)
+
+
+ 3
+x
+
+
+x
+
+
+separation yields 3 equations
+substitution :  
+c11=0
+
+
+                                      2
+eta_y= - 2*log(x)*c8*y - 2*c7*y - c8*x  + c8*y
+
+
+xi_x= - log(x)*c8*x - c7*x
+
+
+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 :  
+c14=0
+
+
+                                      2
+eta_y= - 2*log(x)*c8*y - 2*c7*y - c8*x  + c8*y
+
+
+xi_x= - log(x)*c8*x - c7*x
+
+
+substitution :  
+c15=0
+
+
+                                      2
+eta_y= - 2*log(x)*c8*y - 2*c7*y - c8*x  + c8*y
+
+
+xi_x= - log(x)*c8*x - c7*x
+
+
+substitution :  
+c12=c7
+
+
+                                      2
+eta_y= - 2*log(x)*c8*y - 2*c7*y - c8*x  + c8*y
+
+
+xi_x= - log(x)*c8*x - c7*x
+
+
+substitution :  
+c13=c8
+
+
+                                      2
+eta_y= - 2*log(x)*c8*y - 2*c7*y - c8*x  + c8*y
+
+
+xi_x= - log(x)*c8*x - c7*x
+
+
+substitution :  
+c10=c8
+
+
+                                      2
+eta_y= - 2*log(x)*c8*y - 2*c7*y - c8*x  + c8*y
+
+
+xi_x= - log(x)*c8*x - c7*x
+
+
+substitution :  
+c9=c7
+
+
+                                      2
+eta_y= - 2*log(x)*c8*y - 2*c7*y - c8*x  + c8*y
+
+
+xi_x= - log(x)*c8*x - c7*x
+
+
+End of this CRACK run
+
+The solution : 
+xi_x= - log(x)*c8*x - c7*x
+
+
+                                      2
+eta_y= - 2*log(x)*c8*y - 2*c7*y - c8*x  + c8*y
+
+Free functions or constants : c7  c8  
+**************************************************************************
+
+
+CRACK needed :  2080 ms    GC time : 250 ms
+
+
+Remaining free functions after the last CRACK-run:
+c7  c8  
+
+
+The symmetries are:
+
+xi_x= - log(x)*c8*x - c7*x
+
+                                      2
+eta_y= - 2*log(x)*c8*y - 2*c7*y - c8*x  + c8*y
+
+with c7  c8  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)  
+
+
+time to formulate conditions: 170 ms    GC time : 0 ms
+
+This is CRACK - a solver for overdetermined partial differential equations
+Version 1995-03-20
+**************************************************************************
+
+equations: 
+0= - 2*df(xi_y,u)*u|1 - 2*df(xi_y,x)
+
+
+functions: eta_u(u,y,x)  xi_y(u,y,x)  xi_x(u,y,x)  
+separation w.r.t. u|1  
+new function: c18(u,y)  
+new function: c19(u,y)  
+integrated equation :  
+0=c19 + xi_y
+
+
+new function: c20(y,x)  
+new function: c21(y,x)  
+integrated equation :  
+0=c21 + xi_y
+
+
+separation yields 2 equations
+substitution :  
+xi_y= - c21
+
+eta_u(u,y,x)  
+xi_y= - c21
+
+xi_x(u,y,x)  
+generalized separation 
+new function: c22(y)  
+new function: c23(y)  
+separation yields 3 equations
+substitution :  
+c21=c22
+
+eta_u(u,y,x)  
+xi_y= - c22
+
+xi_x(u,y,x)  
+substitution :  
+c19=c22
+
+eta_u(u,y,x)  
+xi_y= - c22
+
+xi_x(u,y,x)  
+substitution :  
+c22= - c23
+
+eta_u(u,y,x)  
+xi_y=c23
+
+xi_x(u,y,x)  
+End of this CRACK run
+
+The solution : 
+xi_y=c23
+
+Free functions or constants : xi_x(u,y,x)  eta_u(u,y,x)  c23(y)  
+**************************************************************************
+
+
+CRACK needed :  120 ms    GC time : 0 ms
+
+
+Remaining free functions after the last CRACK-run:
+xi_x(u,y,x)  eta_u(u,y,x)  c23(y)  
+
+
+time to formulate conditions: 100 ms    GC time : 0 ms
+
+This is CRACK - a solver for overdetermined partial differential equations
+Version 1995-03-20
+**************************************************************************
+
+equations: expr. with 12 terms
+
+functions: xi_x(u,y,x)  eta_u(u,y,x)  c23(y)  
+separation w.r.t. u|1  
+separation w.r.t. u|2  
+separation w.r.t. u|2  
+new function: c24(y,x)  
+new function: c25(y,x)  
+new function: c26(y,x)  
+new function: c27(y,x)  
+integrated equation :  
+0=c26 + c27*u + xi_x
+
+
+new function: c28(y,x)  
+integrated equation :  
+0=df(eta_u,u) - 2*df(xi_x,x) + c28
+
+
+new function: c29(y,x)  
+new function: c30(y,x)  
+integrated equation :  
+0=c30 + xi_x
+
+
+new function: c31(u,y)  
+new function: c32(u,y)  
+integrated equation :  
+0=df(c23,y)*x + c32 - 2*xi_x
+
+
+separation yields 6 equations
+substitution :  
+xi_x= - c30
+
+
+xi_x= - c30
+
+eta_u(u,y,x)  c23(y)  
+generalized separation 
+new function: c33(y)  
+new function: c34(y)  
+separation yields 3 equations
+substitution :  
+c32= - c34
+
+
+xi_x= - c30
+
+eta_u(u,y,x)  c23(y)  
+substitution :  
+      - df(c23,y)*x + c34
+c30=----------------------
+              2
+
+
+      df(c23,y)*x - c34
+xi_x=-------------------
+              2
+
+eta_u(u,y,x)  c23(y)  
+substitution :  
+c33= - c34
+
+
+      df(c23,y)*x - c34
+xi_x=-------------------
+              2
+
+eta_u(u,y,x)  c23(y)  
+new function: c35(x,y)  
+new function: c36(x,y)  
+integrated equation :  
+0=df(c23,y)*u - c28*u + c36 - eta_u
+
+
+substitution :  
+eta_u=df(c23,y)*u - c28*u + c36
+
+
+      df(c23,y)*x - c34
+xi_x=-------------------
+              2
+
+
+eta_u=df(c23,y)*u - c28*u + c36
+
+c23(y)  
+new function: c37(y)  
+new function: c38(y)  
+integrated equation :  
+               2
+0=df(c23,y,2)*x  - 2*df(c34,y)*x - 8*c28 + 2*c38
+
+
+substitution :  
+                  2
+     df(c23,y,2)*x  - 2*df(c34,y)*x + 2*c38
+c28=----------------------------------------
+                       8
+
+
+      df(c23,y)*x - c34
+xi_x=-------------------
+              2
+
+
+                         2
+        - df(c23,y,2)*u*x  + 8*df(c23,y)*u + 2*df(c34,y)*u*x + 8*c36 - 2*c38*u
+eta_u=-------------------------------------------------------------------------
+                                          8
+
+c23(y)  
+separation w.r.t. u  
+separation yields 2 equations
+substitution :  
+c27=0
+
+
+      df(c23,y)*x - c34
+xi_x=-------------------
+              2
+
+
+                         2
+        - df(c23,y,2)*u*x  + 8*df(c23,y)*u + 2*df(c34,y)*u*x + 8*c36 - 2*c38*u
+eta_u=-------------------------------------------------------------------------
+                                          8
+
+c23(y)  
+separation w.r.t. u  
+separation w.r.t. x  
+new constant: c39
+new constant: c40
+new constant: c41
+new constant: c42
+new constant: c43
+new constant: c44
+integrated equation :  
+                       2
+0=2*c23 + 2*c42 + c43*y  + 2*c44*y
+
+
+new constant: c45
+new constant: c46
+new constant: c47
+new constant: c48
+integrated equation :  
+0=c34 + c47 + c48*y
+
+
+new constant: c49
+integrated equation :  
+0=5*df(c23,y) - c38 + c49
+
+
+separation yields 4 equations
+substitution :  
+c34= - c47 - c48*y
+
+
+      df(c23,y)*x + c47 + c48*y
+xi_x=---------------------------
+                  2
+
+
+                         2
+        - df(c23,y,2)*u*x  + 8*df(c23,y)*u + 8*c36 - 2*c38*u - 2*c48*u*x
+eta_u=-------------------------------------------------------------------
+                                       8
+
+c23(y)  
+substitution :  
+      - df(c23,y)*x - c47 - c48*y
+c26=------------------------------
+                  2
+
+
+      df(c23,y)*x + c47 + c48*y
+xi_x=---------------------------
+                  2
+
+
+                         2
+        - df(c23,y,2)*u*x  + 8*df(c23,y)*u + 8*c36 - 2*c38*u - 2*c48*u*x
+eta_u=-------------------------------------------------------------------
+                                       8
+
+c23(y)  
+substitution :  
+                     2
+      - 2*c42 - c43*y  - 2*c44*y
+c23=-----------------------------
+                  2
+
+
+       - c43*x*y - c44*x + c47 + c48*y
+xi_x=----------------------------------
+                     2
+
+
+                                2
+       8*c36 - 2*c38*u + c43*u*x  - 8*c43*u*y - 8*c44*u - 2*c48*u*x
+eta_u=--------------------------------------------------------------
+                                    8
+
+
+                     2
+      - 2*c42 - c43*y  - 2*c44*y
+c23=-----------------------------
+                  2
+
+
+substitution :  
+c38= - 5*c43*y - 5*c44 + c49
+
+
+       - c43*x*y - c44*x + c47 + c48*y
+xi_x=----------------------------------
+                     2
+
+
+                      2
+       8*c36 + c43*u*x  + 2*c43*u*y + 2*c44*u - 2*c48*u*x - 2*c49*u
+eta_u=--------------------------------------------------------------
+                                    8
+
+
+                     2
+      - 2*c42 - c43*y  - 2*c44*y
+c23=-----------------------------
+                  2
+
+
+decoupling: 
+c36
+
+
+new equations: 
+End of this CRACK run
+
+The solution : 
+                     2
+      - 2*c42 - c43*y  - 2*c44*y
+c23=-----------------------------
+                  2
+
+
+                      2
+       8*c36 + c43*u*x  + 2*c43*u*y + 2*c44*u - 2*c48*u*x - 2*c49*u
+eta_u=--------------------------------------------------------------
+                                    8
+
+
+       - c43*x*y - c44*x + c47 + c48*y
+xi_x=----------------------------------
+                     2
+
+Remaining conditions : 
+0=df(c36,x,2) - df(c36,y)
+
+for the functions : c42  c47  c48  c43  c44  
+c49  c36(x,y)  
+**************************************************************************
+
+
+CRACK needed :  980 ms    GC time : 110 ms
+
+
+Remaining free functions after the last CRACK-run:
+c42  c47  c48  c43  c44  
+c49  c36(x,y)  
+
+
+Free constants and/or functions have been rescaled.
+
+The symmetries are:
+
+xi_x= - 4*c43*x*y - 2*c44*x + c47 + 2*c48*y
+
+                     2
+xi_y= - c42 - 4*c43*y  - 4*c44*y
+
+                   2
+eta_u=c36 + c43*u*x  + 2*c43*u*y + c44*u - c48*u*x - c49*u
+
+with c42  c47  c48  c43  c44  
+c49  c36(x,y)  
+which still have to satisfy: 
+
+0=df(c36,x,2) - df(c36,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 1995-03-20
+**************************************************************************
+
+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: c50(x)  
+integrated equation :  
+   2*x*y
+0=e     *h_2 - c50
+
+
+separation yields 4 equations
+substitution :  
+      c50
+h_2=--------
+      2*x*y
+     e
+
+
+      c50
+h_2=--------
+      2*x*y
+     e
+
+h_1(y,x)  h_0(y,x)  
+substitution :  
+     df(h_0,x)*x
+h_1=-------------
+          2
+         y
+
+
+      c50
+h_2=--------
+      2*x*y
+     e
+
+
+     df(h_0,x)*x
+h_1=-------------
+          2
+         y
+
+h_0(y,x)  
+new function: c51(x)  
+integrated equation :  
+               2    2*x*y            3    x*y      3  2            3          2
+0=df(c50,x)*x*y  - e     *df(h_0,x)*x  + e   *c51*x *y  - 2*c50*x*y  - 6*c50*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          4
+0=e     *df(h_0,x,2)*x  - e     *df(h_0,x)*x + e     *df(h_0,y)*x*y  - 2*c50*y
+
+
+               2    2*x*y            3    x*y      3  2            3          2
+0=df(c50,x)*x*y  - e     *df(h_0,x)*x  + e   *c51*x *y  - 2*c50*x*y  - 6*c50*y
+
+
+functions: 
+      c50
+h_2=--------
+      2*x*y
+     e
+
+
+     df(h_0,x)*x
+h_1=-------------
+          2
+         y
+
+h_0(y,x)  c51(x)  c50(x)  
+separation w.r.t. y  
+linear independent expressions : 
+1
+
+
+y
+
+
+ 2
+y
+
+
+ x*y
+e
+
+
+   x*y
+y*e
+
+
+new constant: c52
+integrated equation :  
+0=c51 - c52*x
+
+
+new constant: c53
+new constant: c54
+integrated equation :  
+0=c51 + c53 + c54*x
+
+
+new constant: c55
+integrated equation :  
+             4
+0=c50 - c55*x
+
+
+new constant: c56
+new constant: c57
+integrated equation :  
+              4              4
+0=log(x)*c57*x  - c50 + c56*x
+
+
+new constant: c58
+new constant: c59
+new constant: c60
+integrated equation :  
+             6        4        3
+0=c50 - c58*x  - c59*x  - c60*x
+
+
+separation yields 5 equations
+substitution :  
+c51=c52*x
+
+
+      c50
+h_2=--------
+      2*x*y
+     e
+
+
+     df(h_0,x)*x
+h_1=-------------
+          2
+         y
+
+h_0(y,x)  
+separation w.r.t. x  
+separation yields 2 equations
+substitution :  
+c53=0
+
+
+      c50
+h_2=--------
+      2*x*y
+     e
+
+
+     df(h_0,x)*x
+h_1=-------------
+          2
+         y
+
+h_0(y,x)  
+substitution :  
+         4
+c50=c55*x
+
+
+          4
+     c55*x
+h_2=--------
+      2*x*y
+     e
+
+
+     df(h_0,x)*x
+h_1=-------------
+          2
+         y
+
+h_0(y,x)  
+separation w.r.t. x  
+linear independent expressions : 
+ 4
+x *log(x)
+
+
+ 4
+x
+
+
+separation yields 2 equations
+substitution :  
+c57=0
+
+
+          4
+     c55*x
+h_2=--------
+      2*x*y
+     e
+
+
+     df(h_0,x)*x
+h_1=-------------
+          2
+         y
+
+h_0(y,x)  
+new function: c61(y)  
+substitution :  
+c56=c55
+
+
+          4
+     c55*x
+h_2=--------
+      2*x*y
+     e
+
+
+     df(h_0,x)*x
+h_1=-------------
+          2
+         y
+
+h_0(y,x)  
+substitution :  
+c54= - c52
+
+
+          4
+     c55*x
+h_2=--------
+      2*x*y
+     e
+
+
+     df(h_0,x)*x
+h_1=-------------
+          2
+         y
+
+h_0(y,x)  
+separation w.r.t. x  
+separation yields 3 equations
+substitution :  
+c60=0
+
+
+          4
+     c55*x
+h_2=--------
+      2*x*y
+     e
+
+
+     df(h_0,x)*x
+h_1=-------------
+          2
+         y
+
+h_0(y,x)  
+substitution :  
+c58=0
+
+
+          4
+     c55*x
+h_2=--------
+      2*x*y
+     e
+
+
+     df(h_0,x)*x
+h_1=-------------
+          2
+         y
+
+h_0(y,x)  
+substitution :  
+c59=c55
+
+
+          4
+     c55*x
+h_2=--------
+      2*x*y
+     e
+
+
+     df(h_0,x)*x
+h_1=-------------
+          2
+         y
+
+h_0(y,x)  
+decoupling: 
+h_0
+
+
+new equations: 
+new function: c62(y)  
+new function: c63(y)  
+new function: c64(y)  
+new function: c65(y)  
+integrated equation :  expr. with 10 terms
+
+substitution :  
+      2*sqrt(2)*x*y + 2*x*y                  sqrt(2)*x*y + 2*x*y
+h_0=(e                     *sqrt(2)*c62 - 2*e                   *c65*y
+
+           sqrt(2)*x*y + x*y        2      sqrt(2)*x*y + x*y
+      - 2*e                 *c52*x*y  - 2*e                 *c52*y
+
+           sqrt(2)*x*y      2  3      sqrt(2)*x*y        2
+      + 2*e           *c55*x *y  + 4*e           *c55*x*y
+
+           sqrt(2)*x*y          2*x*y                  sqrt(2)*x*y + 2*x*y
+      + 2*e           *c55*y - e     *sqrt(2)*c63)/(2*e                   *y)
+
+
+          4
+     c55*x
+h_2=--------
+      2*x*y
+     e
+
+
+      2*sqrt(2)*x*y + 2*x*y          sqrt(2)*x*y + x*y      2  2
+h_1=(e                     *c62*x + e                 *c52*x *y
+
+           sqrt(2)*x*y      3  3      sqrt(2)*x*y      2  2    2*x*y
+      - 2*e           *c55*x *y  - 2*e           *c55*x *y  + e     *c63*x)/(
+
+    sqrt(2)*x*y + 2*x*y  2
+   e                   *y )
+
+
+h_0= expr. with 9 terms
+
+equations: 
+    2*sqrt(2)*x*y + 2*x*y        2*x*y
+   e                     *c62 + e     *c63
+0=-----------------------------------------
+                 sqrt(2)*x*y
+                e
+
+expr. with 12 terms
+
+functions: 
+          4
+     c55*x
+h_2=--------
+      2*x*y
+     e
+
+
+      2*sqrt(2)*x*y + 2*x*y          sqrt(2)*x*y + x*y      2  2
+h_1=(e                     *c62*x + e                 *c52*x *y
+
+           sqrt(2)*x*y      3  3      sqrt(2)*x*y      2  2    2*x*y
+      - 2*e           *c55*x *y  - 2*e           *c55*x *y  + e     *c63*x)/(
+
+    sqrt(2)*x*y + 2*x*y  2
+   e                   *y )
+
+
+h_0= expr. with 9 terms
+c62(y)  c63(y)  c65(y)  
+separation w.r.t. x  
+linear independent expressions : 
+ 2*x*y + 2*sqrt(2)*x*y
+e
+
+
+ 2*x*y
+e
+
+
+separation yields 2 equations
+substitution :  
+c62=0
+
+
+          4
+     c55*x
+h_2=--------
+      2*x*y
+     e
+
+
+      sqrt(2)*x*y + x*y      2  2      sqrt(2)*x*y      3  3
+h_1=(e                 *c52*x *y  - 2*e           *c55*x *y
+
+           sqrt(2)*x*y      2  2    2*x*y          sqrt(2)*x*y + 2*x*y  2
+      - 2*e           *c55*x *y  + e     *c63*x)/(e                   *y )
+
+
+           sqrt(2)*x*y + 2*x*y            sqrt(2)*x*y + x*y        2
+h_0=( - 2*e                   *c65*y - 2*e                 *c52*x*y
+
+           sqrt(2)*x*y + x*y            sqrt(2)*x*y      2  3
+      - 2*e                 *c52*y + 2*e           *c55*x *y
+
+           sqrt(2)*x*y        2      sqrt(2)*x*y          2*x*y
+      + 4*e           *c55*x*y  + 2*e           *c55*y - e     *sqrt(2)*c63)/(2
+
+     sqrt(2)*x*y + 2*x*y
+   *e                   *y)
+
+
+substitution :  
+c63=0
+
+
+          4
+     c55*x
+h_2=--------
+      2*x*y
+     e
+
+
+      x*y      2          3            2
+     e   *c52*x  - 2*c55*x *y - 2*c55*x
+h_1=-------------------------------------
+                    2*x*y
+                   e
+
+
+         2*x*y        x*y            x*y            2  2
+      - e     *c65 - e   *c52*x*y - e   *c52 + c55*x *y  + 2*c55*x*y + c55
+h_0=-----------------------------------------------------------------------
+                                     2*x*y
+                                    e
+
+
+new constant: c66
+integrated equation :  
+0=c65 + c66
+
+
+substitution :  
+c65= - c66
+
+
+          4
+     c55*x
+h_2=--------
+      2*x*y
+     e
+
+
+      x*y      2          3            2
+     e   *c52*x  - 2*c55*x *y - 2*c55*x
+h_1=-------------------------------------
+                    2*x*y
+                   e
+
+
+      2*x*y        x*y            x*y            2  2
+     e     *c66 - e   *c52*x*y - e   *c52 + c55*x *y  + 2*c55*x*y + c55
+h_0=--------------------------------------------------------------------
+                                    2*x*y
+                                   e
+
+
+End of this CRACK run
+
+The solution : 
+      2*x*y        x*y            x*y            2  2
+     e     *c66 - e   *c52*x*y - e   *c52 + c55*x *y  + 2*c55*x*y + c55
+h_0=--------------------------------------------------------------------
+                                    2*x*y
+                                   e
+
+
+      x*y      2          3            2
+     e   *c52*x  - 2*c55*x *y - 2*c55*x
+h_1=-------------------------------------
+                    2*x*y
+                   e
+
+
+          4
+     c55*x
+h_2=--------
+      2*x*y
+     e
+
+Free functions or constants : c52  c55  c66  
+**************************************************************************
+
+
+CRACK needed :  8530 ms    GC time : 570 ms
+                              2      4    x*y              2
+A first integral is:  (df(y,x) *c55*x  + e   *df(y,x)*c52*x
+
+                     3                    2    x*y            x*y
+    - 2*df(y,x)*c55*x *y - 2*df(y,x)*c55*x  - e   *c52*x*y - e   *c52
+
+           2  2                     2*x*y
+    + c55*x *y  + 2*c55*x*y + c55)/e
+
+and an integrating factor:  
+
+  2                 2    x*y
+ x *(2*df(y,x)*c55*x  + e   *c52 - 2*c55*x*y - 2*c55)
+------------------------------------------------------
+                         2*x*y
+                        e
+
+free constants: c52  c55  
+..........................................................................
+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 1995-03-20
+**************************************************************************
+
+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: c67(y)  
+new function: c68(y)  
+integrated equation :  
+0=c68 + u_
+
+
+new function: c69(x)  
+new function: c70(x)  
+integrated equation :  
+0=c70 + u_
+
+
+separation yields 3 equations
+substitution :  
+u_= - c70
+
+
+u_= - c70
+
+v_(y,x)  
+generalized separation 
+new constant: c71
+new constant: c72
+separation yields 3 equations
+substitution :  
+c68= - c71
+
+
+u_= - c70
+
+v_(y,x)  
+substitution :  
+c70= - c71
+
+
+u_=c71
+
+v_(y,x)  
+substitution :  
+c71= - c72
+
+
+u_= - c72
+
+v_(y,x)  
+new function: c73(x)  
+new function: c74(x)  
+integrated equation :  
+                     3
+0=2*c72*x*y + 4*c72*y  + c74 + v_
+
+
+substitution :  
+                         3
+v_= - 2*c72*x*y - 4*c72*y  - c74
+
+
+u_= - c72
+
+
+                         3
+v_= - 2*c72*x*y - 4*c72*y  - c74
+
+
+End of this CRACK run
+
+The solution : 
+                         3
+v_= - 2*c72*x*y - 4*c72*y  - c74
+
+
+u_= - c72
+
+Free functions or constants : c74(x)  c72  
+**************************************************************************
+
+
+CRACK needed :  240 ms    GC time : 0 ms
+                                 2             2
+The solution:  L =  - (2*(x + 2*y )*y + df(y,x) )
+..........................................................................
+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 1995-03-20
+**************************************************************************
+
+equations: 
+            2                                2     2               2
+0=df(a#,x)*y  + df(b#,x)*y - a#*b#*y + a#*f*y  - b#  + b#*f*y + q*y
+
+
+functions: a#(x)  b#(x)  
+separation w.r.t. y  
+new constant: c75
+integrated equation :  
+   int(f,x)           int(f,x)
+0=e        *a# + int(e        *q,x) - c75
+
+
+separation yields 3 equations
+substitution :  
+b#=0
+
+a#(x)  
+b#=0
+
+
+substitution :  
+            int(f,x)
+     - int(e        *q,x) + c75
+a#=-----------------------------
+              int(f,x)
+             e
+
+
+            int(f,x)
+     - int(e        *q,x) + c75
+a#=-----------------------------
+              int(f,x)
+             e
+
+
+b#=0
+
+
+End of this CRACK run
+
+The solution : 
+b#=0
+
+
+            int(f,x)
+     - int(e        *q,x) + c75
+a#=-----------------------------
+              int(f,x)
+             e
+
+Free functions or constants : c75  
+**************************************************************************
+
+
+CRACK needed :  530 ms    GC time : 0 ms
+                            int(f,x)
+                     int(1/e        ,x)*c75
+                    e                      *c76
+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
+(TIME:  crack 23309 24969)