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