Past exam of the mathematical methods I course of the University of Oxford / 2014 / 5 / Solution

Rearrange as $x^{\prime \prime }-4x^{\prime }+3x=10t$. The characteristic polynomial factors as $(r-1)(r-3)$, giving the homogeneous solutions $e^t,e^{3t}$. A particular polynomial solution $At+B$ obeys $3A=10$ and $3B-4A=0$. HenceThis is the full two-constant solution of the second-order linear differential equation.