 Solution

The characteristic polynomial of the homogeneous constant-coefficient equation is $r^2+2r+2$, with roots $-1\pm i$. Its real complementary function is therefore $e^{-x}(C_1\cos x+C_2\sin x)$. The forcing $e^{-x}$ is not resonant because $r=-1$ is not a root. Substituting $y_p=A{e}^{-x}$ into the differential operator gives $A{e}^{-x}$, so $A=1$. HenceA useful independent check is the substitution $y=e^{-x}v$: the equation becomes $v^{\prime \prime }+v=1$, whose general solution is $v=C_1\cos x+C_2\sin x+1$.