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)