The given differential equation is:

$(D^2 - 3D + 2)y = \sin 3x + x^2 + x + e^{4x}$

Here, $D$ is the differential operator, where $D = \frac{d}{dx}$. This means the equation can be expanded as follows:

$\left( \frac{d^2}{dx^2} - 3\frac{d}{dx} + 2 \right)y = \sin 3x + x^2 + x + e^{4x}$

Solution Outline:

1. Find the complementary solution $y_c$ by solving the homogeneous equation: $(D^2 - 3D + 2)y = 0$
2. Find the particular solution $y_p$ by applying the method of undetermined coefficients to the right-hand side.

Step 1: Complementary Solution

The characteristic equation for the homogeneous part is:

$D^2 - 3D + 2 = 0$

Solving for $D$: $D = 1 \quad \text{or} \quad D = 2$

Thus, the complementary solution is: $y_c = c_1 e^x + c_2 e^{2x}$ where $c_1$ and $c_2$ are constants.

Step 2: Particular Solution

For the particular solution, we consider the terms on the right-hand side $\sin 3x + x^2 + x + e^{4x}$. We solve each part individually.

1. For $\sin 3x$: Since there are no roots of the characteristic equation corresponding to $D = 3i$, we assume the particular solution is of the form: $y_p^{(1)} = A \cos 3x + B \sin 3x$

2. For $x^2 + x$: We take a polynomial of the same degree: $y_p^{(2)} = Cx^2 + Dx + E$

3. For $e^{4x}$: Since $D = 4$ is not a root of the characteristic equation, we assume: $y_p^{(3)} = Fe^{4x}$

Step 3: General Solution

The general solution is the sum of the complementary and particular solutions: $y = y_c + y_p = c_1 e^x + c_2 e^{2x} + A \cos 3x + B \sin 3x + Cx^2 + Dx + E + Fe^{4x}$

Would you like me to proceed by calculating the coefficients for the particular solution in detail?

Here are some related questions you could explore:

1. How do you apply the method of undetermined coefficients?
2. What is the role of the characteristic equation in solving linear differential equations?
3. How do you handle repeated roots in the characteristic equation?
4. Can variation of parameters be used in this problem?
5. How do solutions differ when the right-hand side has terms similar to the complementary solution?

Tip: When solving non-homogeneous differential equations, always check if the form of the right-hand side matches the solution to the homogeneous part.