Differential equation solution using Undetermining Coefficients


Solution of differential equation

$y”+8y’+32y=8x^{3}e^{-4x}sin(4x)$

Using method of undetermining coefficients

What i try:

Auxiliary equation corrosponding to that equation is

$$r^2+8r+32=0Longrightarrow r=frac{-8pmsqrt{64-128}}{2}=frac{-8pm 8i}{2}=-4pm 4i$$

So $$y_{c}=A_{1}e^{(-4+4i)x}+A_{2}e^{(-4-4i)x}=c_{1}e^{-4x}cos(4x)+c_{2}e^{-4x}sin(4x)$$

But i did not understand How do i find perticular solution using method of undetermining Coefficients.

Please Help me. Thanks