 Solution

The homogeneous characteristic equation is $r^2-r-2=(r-2)(r+1)=0$, so its solutions are $e^{2x}$ and $e^{-x}$. Because the forcing contains the latter exponential, set $y=e^{-x}v$. Direct differentiation reduces the inhomogeneous linear differential equation toA polynomial particular solution is $v=-3x^2-2x$, since $v^{\prime \prime }-3v^{\prime }=-6-3(-6x-2)=18x$. ThereforeThe exponentially growing first term cannot be canceled by the decaying second term, so boundedness at positive infinity requires $C_1=0$. The initial condition $y(0)=1$ sets $C_2=1$. Hence