The elimination of the arbitrary constants A, Band C from $y=A+Bx+C{e}^{-x}$ leads to the differential equation:

Given $y=A+Bx+C{e}^{-x}$       ... (i)

$\begin{array}{l}\Rightarrow \frac{dy}{dx}=B-C{e}^{-x } ... \left(\mathrm{ii}\right)\\ \Rightarrow \frac{d^{2}y}{d{x}^2}=C{e}^{-x} ... \left(\mathrm{iii}\right)\end{array}$

and $\frac{d^{3}y}{d{x}^3}=-C{e}^{-x}$      ... (iv)

Adding (iii) and (iv) we get, $\frac{d^{3}y}{d{x}^3}+\frac{d^{2}y}{d{x}^2}=0$

i.e., $y^{\text{'}\text{'}\text{'}}+y^{\text{'}}=0$