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You should use the method substitute_function instead of subs and use a symbol different from p to denote p(r) (otherwise there is a confusion between the function p and its value at r). Besides the semicolons at the end of each line are not necessary in Python and the line v = var('v') is useless, since the Python variable v is redeclared in the next line. So basically, your code should be:

sage: r = var('r')
sage: P = function('p')(r)
sage: v = P + P.derivative(r)
sage: p1(r) = desolve(P.derivative(r)==0, P, ivar=r); p1(r)
_C
sage: v.substitute_function(p, p1)
_C