 Solution

The characteristic equation is $m^2+3m+2=(m+1)(m+2)=0$, giving the complementary function $C_1{e}^{-x}+C_2{e}^{-2x}$. For the forcing term $x$, a linear particular solution $ax+b$ gives $2ax+3a+2b=x$, hence $a=1/2$, $b=-3/4$.

The forcing $-e^{-x}$ gives resonance in a differential equation, since $e^{-x}$ already appears in the complementary function. Try $Kx{e}^{-x}$. Direct differentiation givesso $K=-1$. Combining the two particular solutions givesThe factor $x$ in the resonant exponential particular solution is required because $-1$ is already a root of the characteristic equation.