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

Given $y=A+Bx-C{e}^{-x}$

$\Rightarrow y^{\mathrm{\prime }}=B+C\cdot e^{-x}$                   (i)

and ${\mathrm{y}}^{\prime \prime }=-{\mathrm{Ce}}^{-\mathrm{x}}$                          (ii)

and ${\mathrm{y}}^{\prime \prime \mathrm{\prime }}={\mathrm{Ce}}^{-\mathrm{x}}$                             (iii)

From eqs. (ii) and (iii), 

$y^{\prime \prime \mathrm{\prime }}=-y^{\prime \prime }\Rightarrow y^{\prime \prime \mathrm{\prime }}+y^{\prime \prime }=0$