Past exam of the mathematical methods I course of the University of Oxford / 2018 / 8 / a / Solution

The characteristic equation $r^2+5r+4=(r+1)(r+4)=0$ gives the complementary function $C_1{e}^{-x}+C_2{e}^{-4x}$. Write the forcing as $30e^x+30e^{-x}$. The method of undetermined coefficients gives $3e^x$ for the first term. Because $e^{-x}$ already belongs to the homogeneous solution, resonance in a differential equation requires a trial $Kx{e}^{-x}$ for the second; substitution gives $3K=30$. Therefore