Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2006 / Paper 1 / 10 / c

The differential operator is $(D-1{)}^2$, where $D=d/dx$. Set $y=e^{x}v$; the product rule reduces the equation to $v^{\prime \prime }=1+x$. Twice integrating gives $v=x^2/2+x^3/6+Ax+B$. The initial conditions yield $B=A=0$, soThis substitution removes the repeated characteristic root before finding a particular solution.