https://file.aiquickdraw.com/mathbot/file/1726330479576vfrxr96n.jpg

The equation provided in the image is:

\[
(D^2 + D + 1)y = e^{2x} + x^2 + \sin(2x)
\]

Where \(D\) represents the differential operator, meaning \(D = \frac{d}{dx}\). This is a second-order linear non-homogeneous differential equation.

We will solve this equation in two steps:
1. **Solve the homogeneous equation**: \( (D^2 + D + 1)y_h = 0 \)
2. **Find a particular solution** \( y_p \) for the non-homogeneous part: \( (D^2 + D + 1)y_p = e^{2x} + x^2 + \sin(2x) \)

### Step 1: Solving the Homogeneous Equation
The homogeneous equation is:

\[
(D^2 + D + 1)y_h = 0
\]

This corresponds to the characteristic equation:

\[
r^2 + r + 1 = 0
\]

Solving this quadratic equation using the quadratic formula:

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

The roots are:

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

These are complex roots, so the solution to the homogeneous equation is:

\[
y_h(x) = e^{-\frac{1}{2}x}\left( C_1 \cos\left( \frac{\sqrt{3}}{2}x \right) + C_2 \sin\left( \frac{\sqrt{3}}{2}x \right) \right)
\]

### Step 2: Finding the Particular Solution
Now, we find a particular solution \(y_p\) for the non-homogeneous part:

\[
(D^2 + D + 1)y_p = e^{2x} + x^2 + \sin(2x)
\]

We solve this by finding particular solutions for each term on the right-hand side separately:

- For \( e^{2x} \), try \( y_p^{(1)} = Ae^{2x} \).
- For \( x^2 \), try \( y_p^{(2)} = Bx^2 + Cx + D \).
- For \( \sin(2x) \), try \( y_p^{(3)} = E\sin(2x) + F\cos(2x) \).

I will now compute the particular solutions step by step.The general solution to the given differential equation is:

\[
y(x) = x^2 - 2x + \left( C_1 \sin\left( \frac{\sqrt{3}}{2}x \right) + C_2 \cos\left( \frac{\sqrt{3}}{2}x \right) \right) e^{-\frac{x}{2}} + \frac{e^{2x}}{7} - \frac{3}{13} \sin(2x) - \frac{2}{13} \cos(2x)
\]

### Explanation of the solution components:
- \( x^2 - 2x \): Comes from the particular solution for the \( x^2 \) term on the right-hand side.
- \( \left( C_1 \sin\left( \frac{\sqrt{3}}{2}x \right) + C_2 \cos\left( \frac{\sqrt{3}}{2}x \right) \right) e^{-\frac{x}{2}} \): Represents the solution to the homogeneous equation.
- \( \frac{e^{2x}}{7} \): Is the particular solution for the \( e^{2x} \) term.
- \( -\frac{3}{13} \sin(2x) - \frac{2}{13} \cos(2x) \): Is the particular solution for the \( \sin(2x) \) term.

Would you like more details on any part of this solution?

Here are 5 related questions to consider:
1. How would the solution change if the right-hand side was different, for example \(e^x\) instead of \(e^{2x}\)?
2. What is the significance of the complex roots of the characteristic equation in terms of the behavior of the solution?
3. How do we determine the form of the particular solution for each term in the non-homogeneous equation?
4. What role do the constants \(C_1\) and \(C_2\) play in the final solution?
5. How would we apply initial conditions to find specific values for \(C_1\) and \(C_2\)?

**Tip**: When solving non-homogeneous differential equations, always start by solving the homogeneous part before addressing the particular solution for the non-homogeneous terms.

Learn how to solve the second-order non-homogeneous differential equation (D^2 + D + 1)y = e^{2x} + x^2 + sin(2x) with a step-by-step guide. This solution involves solving the characteristic equation and finding particular solutions for non-homogeneous terms. Suitable for undergraduate students studying calculus and differential equations.

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