The differential equation to solve is the following
$$ -\dfrac{dc_\mathrm{B}}{dt} = k_1\ c_\mathrm{A,0}\ e^{-k_1 t} - k_2\ c_\mathrm{B}$$
This formula expresses the variation in B $(dc_\mathrm{B})$ concentration as a function of time, where $k_1, k_2$ and $c_\mathrm{A,0}$ are constant.
Applying the solution formula
$$c_\mathrm{B} = e^{-\int k_2\ dt} \left[ \left( \int e^{\int k_2\ dt} \times k_1\ c_\mathrm{A,0}\ e^{-k_1 t} \right) + \mathrm{const.} \right]$$
And applying the initial condition
$$c_\mathrm{B}(0) = 0 \rightarrow \mathrm{const.} = \dfrac{k_1\ c_\mathrm{A,0}}{-k_2-k_1}$$
What I get is
$$c_\mathrm{B} = \dfrac{k_1\ c_\mathrm{A,0}}{-k_2-k_1} e^{(k_2-k_1)t}$$
Detailed steps to get my result
The solution (without steps) shown in the teacher's slides is
$$ c_\mathrm{B} = k_1\ c_\mathrm{A,0} \left( \dfrac{e^{-k_1t}}{k_2-k_1} + \dfrac{e^{-k_2t}}{k_1-k_2} \right) $$
Are these two solutions the same solution written differently? Or am I the one who did something wrong?