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sympy.solvers.solve gives a solution similar to what you expected:

from sympy import solvers, Symbol
y1 = Symbol('y1')
y2 = Symbol('y2')
t = Symbol('t')
k1 = Symbol('k1')
k2 = Symbol('k2')
sol = solvers.solve(y1*exp(-t/k1)/k1 - y2*exp(-t/k2)/k2,t)
show(sol[0]._sage_())