T'= k1*exp(k2*t) - k3(T-T0)
determine T(t), assume that T(-infinity)= T0
I assume this is
$\displaystyle \frac{dT}{dt} = k_1e^{k_2t} - k_3(T - T_0)$
$\displaystyle \frac{dT}{dt}= k_1e^{k_2t} - k_3T + k_3T_0$
$\displaystyle \frac{dT}{dt} + k_3T = k_1e^{k_2t} + k_3T_0$
This is first order linear, so now use the Integrating Factor method.