https://file.aiquickdraw.com/mathbot/file/17262772322084435z208.jpg

The image contains two problems regarding differential equations.

### 1. Particular Integral of the Differential Equation:

The equation given is:
\[
(D^2 + D + 1)y = \sin 2x
\]

You are asked to find the **particular integral** for this equation. The answer choices are:

- (a) \( \frac{1}{13} \sin 2x \)
- (b) \( -\frac{1}{13} (2 \cos 2x + 3 \sin 2x) \)
- (c) \( -\frac{1}{13} \cos 2x \)
- (d) \( \frac{1}{13} (2 \cos 2x - 3 \sin 2x) \)

### 2. Complementary Function of the Differential Equation:

The equation is:
\[
(D^2 + 1)y = e^x
\]

You are asked to find the **complementary function** for this differential equation. The answer choices are:

- (a) \( y = (c_1 \cos x + c_2 \sin x) \)
- (b) \( y = c_1 \cosh x + c_2 \sinh x \)
- (c) \( y = c_1 \cos x - c_2 \sin x \)
- (d) \( y = c_1 \cosh x - c_2 \sinh x \)

Would you like help solving these problems or do you want more details about a specific concept involved here?

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**Relative Questions:**

1. What method is used to find the particular integral of non-homogeneous differential equations?
2. How do you find the complementary function of a second-order linear differential equation?
3. What is the difference between hyperbolic and trigonometric functions in differential equations?
4. How can you determine which answer choices apply for different forms of the solution?
5. How does the operator \( D \) relate to derivatives in solving these equations?

**Tip:** When solving non-homogeneous linear differential equations, always solve for the complementary function first, then focus on the particular integral.

The particular integral and complementary function of the given differential equations.

This page covers how to solve the particular integral and complementary function for second-order differential equations, focusing on sin(2x) and exponential functions. The page explains the key concepts of hyperbolic and trigonometric functions in the solution process.

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