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The image you provided contains a set of differential equation solutions. Here are the equations visible in the image:

5. \( y = e^x(c_1 + c_2 x + c_3 x^2) \)

6. \( y = c_1 \sin(5x) + c_2 \cos(5x) \)

7. \( y = e^{-x}(c_1 \cos(3x) + c_2 \sin(3x)) \)

8. \( y = (c_1 + c_2 x)e^x + c_3 e^{2x} \)

Would you like further clarification or solutions for any specific equation?

Here are five relative questions that might expand your understanding:

1. How can we solve a second-order differential equation using the characteristic equation approach?
2. What is the significance of the constants \(c_1\), \(c_2\), and \(c_3\) in the solutions of these equations?
3. What method is used to derive solutions of the form \( e^{kx} \)?
4. How do sine and cosine terms appear in solutions to differential equations with complex roots?
5. Can you explain how particular solutions are derived when non-homogeneous terms are present?

**Tip**: When solving second-order linear differential equations, it's helpful to know whether the equation is homogeneous or non-homogeneous to apply the correct solving method.

The image contains four equations related to solutions of differential equations.

Explore the solutions to differential equations involving exponential functions and trigonometric terms. This guide covers characteristic equations and the methods used to derive these solutions. Suitable for college-level mathematics.

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