Past exam of the mathematical methods I course of the University of Oxford / 2021 resit / 10 / b / Solution

Use the method of undetermined coefficients with the polynomial $f_p=a{t}^3+b{t}^2+ct+d$. Substitution into $f_p^{\prime \prime }+3f_p^{\prime }+2f_p$ givesMatching successive coefficients with the forcing gives $a=1$, $b=-3$, $c=4$ and $d=1$. Hence a particular integral isIndeed $f_p^{\prime }=3t^2-6t+4$ and $f_p^{\prime \prime }=6t-6$ give $f_p^{\prime \prime }+3f_p^{\prime }+2f_p=2t^3+3t^2-4t+8$. The complementary function, if wanted, is $C_1{e}^{-t}+C_2{e}^{-2t}$.