Solve a system of equations for functions

edit

desolve_system doesn't give the original rburings solution (edited later ?) :

sage: t = SR.var("t")
sage: f, g = function("f, g")
sage: e1 = diff(f(t), t) + diff(g(t), t) == 6
sage: e2 = diff(f(t), t) - diff(g(t), t) == 4
sage: Sol = desolve_system([e1, e2], [f(t), g(t)], ivar=t) ; Sol
[f(t) == 5*t + f(0), g(t) == t + g(0)]

BTW :

sage: from sympy import dsolve, sympify
sage: SSol = [u._sage_() for u in dsolve(*map(sympify, ([e1, e2], [f(t), g(t)])))] ; SSol
[f(t) == C1 + 5*t, g(t) == C2 + t]
sage: mathematica("DSolve[{D[f[t],t] + D[g[t],t] == 6, D[f[t],t] - D[g[t],t] == 4}, {f[t], g[t]}, t]")
{{f[t] -> 5*t + C[1], g[t] -> t + C[2]}}

Cut n' paste error ?

Using the original notations :

sage: %cpaste
Pasting code; enter '--' alone on the line to stop or use Ctrl-D.
:var('r')
:f = function('f')(r)
:g = function('g')(r)
:
:eq1 = diff(f,r) + diff(g,r) == 6
:eq2 = diff(f,r) - diff(g,r) == 4
:--
r
sage: desolve_system([eq1,eq2],[f,g])
[f(r) == 5*r + f(0), g(r) == r + g(0)]

One notes that the functions appear only differentiated. They can be substituted by temporary variables, and the resulting system solved for them, then integrated. But you don't even need that here ; using my notations :

sage: (e1+e2)/2
diff(f(t), t) == 5
sage: ((e1+e2)/2).integrate(t)
f(t) == c1 + 5*t
sage: ((e1-e2)/2).integrate(t)
g(t) == c2 + t