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rhermans
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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)

rhermans
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