To find the complementary function (CF) of the differential equation

$(D^3 - 8)y = \sin(2x) + x^2,$

we begin by focusing on the left-hand side operator, $(D^3 - 8)$, and finding the complementary function. The complementary function corresponds to solving the homogeneous equation

$(D^3 - 8)y = 0.$

Step 1: Solve the homogeneous equation

The equation $(D^3 - 8)y = 0$ can be rewritten as:

$D^3 y - 8y = 0,$ or equivalently, $D^3 y = 8y.$

This is a linear differential equation with constant coefficients. The characteristic equation associated with this is:

$r^3 - 8 = 0.$

Factoring:

$r^3 = 8 \quad \Rightarrow \quad r = \sqrt[3]{8} = 2.$

So, the root $r = 2$ has multiplicity 3, meaning the general solution to the homogeneous equation is:

$y_h = C_1 e^{2x} + C_2 x e^{2x} + C_3 x^2 e^{2x},$ where $C_1, C_2, C_3$ are constants determined by initial conditions (if any are given).

Step 2: Conclusion for the Complementary Function (CF)

Thus, the complementary function (CF) of the differential equation is:

$y_h = C_1 e^{2x} + C_2 x e^{2x} + C_3 x^2 e^{2x}.$

Would you like me to proceed with the particular integral (PI) for the non-homogeneous part, or do you need more details on the CF?

Related Questions:

1. How would you solve the non-homogeneous part of this equation?
2. What methods can be used for solving a higher-order linear differential equation with constant coefficients?
3. How does the method of undetermined coefficients apply to this equation?
4. What happens if there are different types of non-homogeneous terms in the equation (e.g., exponential, polynomial)?
5. Can we apply the method of variation of parameters to solve for the particular solution?

Tip: Always factor the characteristic equation carefully when dealing with repeated roots, as this determines the structure of the complementary function.