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The image you uploaded contains a series of differential equations exercises from a math textbook. The exercises involve solving various types of differential equations and some related questions. Below is a quick breakdown:

1. **Exercises 1 to 6**: 
   These are basic differential equations that ask you to solve for \( y(x) \). They involve:
   - #1: \( \frac{dy}{dx} = x^3 + 2x \)
   - #2: \( \frac{dy}{dx} = \frac{3}{x} \)
   - #3: \( \frac{dy}{dx} = 4 \cos 6x \)
   - #4: \( \frac{dy}{dx} = \frac{4}{x^2 - 1} \)
   - #5: \( \frac{dy}{dx} = \frac{2}{x^2 + 4} \)
   - #6: \( \frac{dy}{dx} = \frac{3}{x^2 + x} \)

2. **Exercises 7 to 12**:
   These involve solving initial value problems. You are given a differential equation with an initial condition (values for \( y \) at a specific \( x \)) to solve for \( y \):
   - #7: \( \frac{dy}{dx} = 3e^x \), \( y = 6 \) when \( x = 0 \)
   - #8: \( \frac{dy}{dx} = 4e^{-3x} \), \( y = 2 \) when \( x = 0 \)
   - #9: \( \frac{dy}{dx} = 4y \), \( y = 3 \) when \( x = 0 \)
   - #10: \( \frac{dy}{dx} = -5y \), \( y = 7 \) when \( x = 0 \)
   - #11: \( \frac{dy}{dx} = 4 \sin 2x \), \( y = 2 \) when \( x = \frac{\pi}{2} \)
   - #12: \( \frac{dy}{dx} = x^2 + 3 + e^{2x} \), \( y = -1 \) when \( x = 0 \)

3. **Exercises 13 and 14**:
   These involve proving that specific families of curves can be interpreted as solutions to given differential equations:
   - #13: Prove that the family of circles \( (x + 1)^2 + (y - 3)^2 = c^2 \) can be solutions to the differential equation \( \frac{dy}{dx} = \frac{-(x+1)}{y-3} \).
   - #14: Prove that the family of parabolas \( y = ax^2 \) can be interpreted as two families of solutions for the equation \( \frac{dy}{dx} = \frac{2y}{x} \).

If you'd like, I can solve any of these exercises or explain any specific concepts from them. Would you like me to solve a particular problem or provide more detailed explanations? 

Also, here are 5 related questions to expand your understanding:

1. How do you approach solving first-order linear differential equations?
2. What is the method of separation of variables, and when is it applicable?
3. How do you find the integrating factor for linear differential equations?
4. What is the importance of initial conditions in solving differential equations?
5. How can exact differential equations be identified and solved?

**Tip:** When dealing with initial value problems, always make sure to solve for the constant of integration using the given conditions to ensure the solution satisfies the entire equation!

The image contains a series of differential equation exercises. The exercises involve solving various types of differential equations and related questions, including initial value problems and proving families of curves as solutions to specific equations.

Explore solving differential equations, initial value problems, and proving families of circles and parabolas as solutions to equations. This page covers mathematical concepts like separation of variables and initial value theorems, suitable for high school or undergraduate students.

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