I played around with sage and found some problems with desolve command.
To solve ode diff(y(x),x)+a*y(x)+b*x+c we should first define variables and
functions
x = var('x')
a,b,c=var('a b c')
y=function('y',x)
eq=diff(y,x)+a*y+b*x+c
but than unless it is obvious that the dependent variable is x and
independent is y we should write such command
desolve(eq,y,ivar=x)
which is really annoying. And what is worse - we get a wrong answer
-((a*x - 1)*b*e^(a*x)/a^2 + c*e^(a*x)/a - c)*e^(-a*x)
but the right answer is
-((a*x - 1)*b*e^(a*x)/a^2 + c*e^(a*x)/a - _C1)*e^(-a*x)
where _C1 - arbitrary constant.
Of course, if we don't use c as a variable in equation we will get a right
answer, but another problem will rise. If we have a bunch of ODE and want
their solutions to interact somehow, than we expect different arbitrary
constant for each one. But sage uses one letter c in every solution, which
causes problems.
Actually the first problem is easy to solve. We can correct
devel/sage-main/build/sage/calculus/desolvers.py
First - make dvar variable which represent dependent variable optional by
giving it default value None.
< #64
def desolve(de, dvar, ics=None, ivar=None, show_method=False,
contrib_ode=False):
>
def desolve(de, dvar=None, ics=None, ivar=None, show_method=False,
contrib_ode=False):
Second - add definition of dvar.
dvar - is one of the functions differentiated.
> after #319
if dvar is None
dvars = extract_func_from_diff(de)
if len(dvars) != 1:
raise ValueError, "Unable to determine dependent variable, please
specify."
dvar = dvars.pop()
Finally - change definition of ivar.
ivar - is argument of dvar and it is also in parameter_set of
FDerivativeOperator
< #324-329
elif ivar is None:
ivars = de.variables()
ivars = [t for t in ivars if t is not dvar]
if len(ivars) != 1:
raise ValueError, "Unable to determine independent variable, please
specify."
ivar = ivars[0]
>
elif ivar is None:
ivars = set(dvar.arguments()).intersection(extract_var_from_diff(de))
if len(ivars) != 1:
raise ValueError, "Unable to determine independent variable, please
specify."
ivar = ivars.pop()
where extract_*_from_diff(de) - are blackboxes which return set of functions
or variables in FDerivativeOperators from de. They can be implemented by
using type checking (instanceof) and operator function (I'm not sure is it
possible to solve the problem using wild card and if it is any analog of
Maple type function)
After the changes it will be possible to call desolve(de) in most cases.
At least it is clever to make simple correction
<#325-326
ivars = de.variables()
ivars = [t for t in ivars if t is not dvar]
>
ivars=dvar.arguments()
After that it will be possible to call desolve(de, dvar) if dvar has only
one argument.
The second problem is harder and there are several solution. Anyway we need
to implement new variable generator (like SR.symbol()) Then we can either
change a little Maxima algorithm to be able to use variables generated by
sage as integration constants or just rename all variables in equation, pass
it to Maxima and then translate the result back using generated variables.
--
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