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You can provide a stronger information than assume(-k/a>0), the following seems to work:

sage: assume(a>0)
sage: assume(k<0)
sage: ans = desolve(de, v, ics=[0,0], ivar=t)
sage: ans
-(a*g*k*t*e^(k*log(-m_0)/a) + g*k*m_0*e^(k*log(a*t - m_0)/a) - g*k*m_0*e^(k*log(-m_0)/a) + ((a^2 - a*k)*e^(k*log(a*t - m_0)/a) - (a^2 - a*k)*e^(k*log(-m_0)/a))*v_e)*e^(-k*log(-m_0)/a)/(a*k - k^2)