Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2001 / Paper 2 / 4 / b / Solution

Now the characteristic equation is $(m+1{)}^2=0$, with a repeated root, so the complementary function is $(C_1+C_{2}x)e^{-x}$. For the polynomial forcing, substitute $a{x}^2+bx+c$ to obtainThus $a=1$, $b=-4$, $c=6$. For the forcing $2e^{-x}$, both $e^{-x}$ and $x{e}^{-x}$ already occur in the complementary function; the resonant exponential particular solution therefore starts with $K{x}^2{e}^{-x}$. The identitygives $K=1$. The resulting general solution is