The image shows a set of differential equation solutions written in general forms. Here’s a transcription of the equations:

5. $y = e^x \left( C_1 + C_2 x + C_3 x^2 \right)$

6. $y = C_1 \sin 5x + C_2 \cos 5x$

7. $y = e^{-x} \left( C_1 \cos 3x + C_2 \sin 3x \right)$

8. $y = \left( C_1 + C_2 x \right) e^x + C_3 e^{2x}$

These are all general solutions to second-order linear homogeneous differential equations, typically involving characteristic equations with real and complex roots.

Would you like more details on how these solutions are derived, or would you like me to explain one of these in detail?

Here are some relative questions:

1. How are the characteristic equations related to the solutions of differential equations?
2. What does the term $e^x \left( C_1 + C_2 x + C_3 x^2 \right)$ signify in terms of the nature of the roots?
3. Can you explain why we get sinusoidal terms like $\sin 5x$ and $\cos 5x$ in some solutions?
4. How are exponential functions like $e^x$ and $e^{-x}$ important in solving second-order differential equations?
5. What is the significance of the constants $C_1, C_2,$ and $C_3$ in these general solutions?

Tip: When solving second-order homogeneous differential equations, the nature of the characteristic equation (real, repeated, or complex roots) plays a crucial role in determining the form of the solution.