Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2013 / Paper 2 / 14 / a / Solution

The characteristic equation has roots zero and three, giving $y_h=C_1+C_2{e}^{3x}$. Since the forcing is quadratic and the homogeneous solution contains a constant, use the polynomial trial $y_p=A{x}^3+B{x}^2+Cx$. Substitution givesMatching coefficients gives $A=-1/3$, $B=-1/3$, and $C=-8/9$. Therefore