Find a particular solution for the differential equation by the method of undetermined coefficients. $2y'' - 16y' + 32y = -e^{4x}$ Also, find the general solution of this equation.
The steps I took to solve this problem,
Find the auxiliary equation which is $2m^2-16m+32=0$ for which the roots are $m_1=4$ and $m_2=4$ so $m=4$ of multiplicity 2.
Solve for a general equation of $y_h(x) = C_1e^{4x}x + C_2e^{4x}$
When I try to find a particular solution by taking the derivates of the right hand side, I get \begin{align} y_p &= Ae^{4x}\\ y_p' &= 4Ae^{4x}\\ y_p'' &= 16Ae^{4x} \end{align} Substituting these values into the left hand side results in $0 = -e^{4x}$ which is not possible. Can someone identify what I am missing?