Calculus 8th Edition

$y_p(x)=x(A x^3+Bx^+Cx+D)e^{x}$
Consider $G(x)=e^{\alpha x} A(x) \sin mx$ or $G(x)=e^{\beta x} A(x) \cos mx$ The trial solution for the method of undetermined coefficients is defined as: $y_p(x)=e^{\alpha x} B(x) \sin mx +e^{\beta x} C(x) \cos mx$ When the sum of the coefficients of a differential equation is zero. Then, we have $y_p(x)=x e^{\alpha x} B(x)$ On comparing the above equation with the given equation, we get the sum of the coefficients of a differential equation is $1+3-4=0$. Thus, the trial solution for the method of undetermined coefficients is: $y_p(x)=x(A x^3+Bx^+Cx+D)e^{x}$