sol = x[t] /. First@DSolve[
{
D[x[t], t] == -k1 x[t] + (1 - x[t]) k2 Exp[-k2 t]
, x[0] == 0
}
, x[t], t];
The Series
2nd order coefficient:
SeriesCoefficient[sol, {t, 0, 2}]
-((k1 k2)/2) - k2^2
Third order
SeriesCoefficient[sol, {t, 0, 3}]
1/6 (k1^2 k2 + 3 k1 k2^2 + 5 k2^3)