Identify each of the differential equations as type (for example, separable, linear first order, linear second order, etc.), and then solve it. $$(D-2)^{2}\left(D^{2}+9\right) y=0$$

When dealing with linear homogeneous differential equations with constant coefficients, the characteristic equation plays a crucial role. Essentially, it translates the differential equation into an algebraic equation that is easier to solve.
The characteristic equation is obtained by replacing the differential operator with a polynomial in terms of a variable, typically denoted as 'r'. For instance, for the given differential equation \[ (D-2)^2 \big(D^2 + 9\big) y = 0 \], the characteristic equation becomes \[ (r-2)^2 (r^2 + 9) = 0 \].
Finding the roots of this algebraic equation helps us determine the form of the general solution.

Linear differential equations with constant coefficients have terms that are easily manageable, leading to straightforward techniques for finding solutions. In the case mentioned, the coefficients (the numbers in front of the differential operators) do not change with respect to the independent variable (usually denoted as 'x').
For example, the differential operator \[ (D-2)^2 \big(D^2 + 9\big) \] has constant coefficients: -2 and 9. These fixed values simplify the process of finding the characteristic equation and solving for its roots, which are essential steps in determining the general solution.
The stability offered by constant coefficients ensures that the methods used for solving these types of differential equations are systematic and reliable.

Once the characteristic equation is solved, and the roots are found, we can construct the general solution to the differential equation. This solution encapsulates all possible specific solutions to the equation, incorporating arbitrary constants to account for initial conditions.
For example, for the characteristic equation \[ (r-2)^2 (r^2 + 9) = 0 \] we have roots: r = 2 (a double root), and \[ r = 3i \text{ and } r = -3i \]. These roots correspond to different parts of the general solution:

1. **Real Roots:** For each real root, we include a term of the form \[ e^{rx} \]. If the root is repeated, we multiply the exponential term by x for each repetition. For our double root at r = 2, the terms are \[ (C_1 + C_2 x) e^{2x} \].
2. **Complex Roots:** For complex roots of the form \[a ± bi \], the solutions involve both sine and cosine functions: \[ e^{ax}(C_3 \text{cos}(bx) + C_4 \text{sin}(bx)) \]. For our roots \[±3i \], the terms are \[ C_3 \text{cos}(3x) + C_4 \text{sin}(3x) \].

Hence, the complete general solution for the given differential equation is \[ y(x) = (C_1 + C_2 x)e^{2x} + C_3 \text{cos}(3x) + C_4 \text{sin}(3x) \], where \[ C_1, C_2, C_3, \] and \[ C_4 \] are arbitrary constants.