To find a particular solution $y_p$ of the differential equation

$y'' + 9y = 2 \cos 3x + 3 \sin 3x$

using the Method of Undetermined Coefficients, we can follow these steps:

1. Identify the form of the particular solution: Since the non-homogeneous term on the right side, $2 \cos 3x + 3 \sin 3x$, is a combination of sine and cosine functions with the same frequency (3), we assume a particular solution of the form:

   $y_p = A \cos 3x + B \sin 3x$

   where $A$ and $B$ are constants to be determined.

2. Calculate derivatives of $y_p$: First derivative: $y_p' = -3A \sin 3x + 3B \cos 3x$

   Second derivative: $y_p'' = -9A \cos 3x - 9B \sin 3x$

3. Substitute into the differential equation: Substitute $y_p$ and $y_p''$ into the left side of the original equation:

   $y_p'' + 9y_p = (-9A \cos 3x - 9B \sin 3x) + 9(A \cos 3x + B \sin 3x)$

   Simplifying this expression gives: $y_p'' + 9y_p = (0) \cos 3x + (0) \sin 3x$