To find the complementary function (CF) for the given differential equation:

$(D^2 + D + 1)y = 1 + x^2 + x^3$

where $D$ denotes the differential operator, we first solve the homogeneous part of the equation:

$(D^2 + D + 1)y = 0$

Step 1: Solve the homogeneous equation

The equation corresponding to the homogeneous part is:

$D^2 + D + 1 = 0$

We need to solve the characteristic equation:

$r^2 + r + 1 = 0$

To find the roots, we use the quadratic formula:

$r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

where $a = 1$, $b = 1$, and $c = 1$:

$r = \frac{-1 \pm \sqrt{1^2 - 4(1)(1)}}{2(1)} = \frac{-1 \pm \sqrt{1 - 4}}{2} = \frac{-1 \pm \sqrt{-3}}{2}$

$r = \frac{-1 \pm i\sqrt{3}}{2}$

Thus, the roots are complex:

$r = -\frac{1}{2} \pm \frac{\sqrt{3}}{2}i$

Step 2: Write the complementary function (CF)

The complementary function (CF) is given by:

$y_c = e^{\alpha x} \left( C_1 \cos(\beta x) + C_2 \sin(\beta x) \right)$

where $\alpha = -\frac{1}{2}$ and $\beta = \frac{\sqrt{3}}{2}$. Therefore, the complementary function is:

$y_c = e^{-\frac{x}{2}} \left( C_1 \cos\left(\frac{\sqrt{3}}{2}x\right) + C_2 \sin\left(\frac{\sqrt{3}}{2}x\right) \right)$

This is the general solution to the homogeneous equation.

Step 3: Non-homogeneous part

To find the particular solution, we would need to solve for the non-homogeneous part $1 + x^2 + x^3$, but for now, we only need the complementary function.

Let me know if you'd like to proceed with solving for the particular solution or if you have further questions!

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Related questions:

1. What is the method to find the particular solution of a non-homogeneous differential equation?
2. How do you use the method of undetermined coefficients to solve non-homogeneous equations?
3. Can you explain how to apply the quadratic formula for complex roots?
4. What is the significance of the complementary function in solving differential equations?
5. How do you handle repeated roots in the characteristic equation?

Tip: When solving for the roots of the characteristic equation, always check for the discriminant to determine whether the roots are real or complex. This affects the form of the complementary function!