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The image shows four differential equations along with their proposed solutions, and the task is to prove that the given functions are indeed solutions of the corresponding differential equations over specified intervals. Here is a detailed approach for each sub-problem:

### a) \( y' + 4xy - 8x = 0 \)
Proposed solution: \( y(x) = ce^{-2x^2} + 2 \), where \( c \in \mathbb{R} \) and the interval is \( I = (-\infty, \infty) \).

**Steps:**
1. Differentiate \( y(x) \) with respect to \( x \) to find \( y'(x) \).
2. Substitute \( y(x) \) and \( y'(x) \) into the differential equation.
3. Simplify to verify that both sides of the equation are equal.

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### b) \( y' = xy^2 \)
Proposed solution: \( y(x) = \frac{2}{c - x^2} \), where \( c \in \mathbb{R}^+ \) and the interval is \( I = (-\sqrt{c}, \sqrt{c}) \).

**Steps:**
1. Differentiate \( y(x) \) to find \( y'(x) \).
2. Substitute \( y(x) \) and \( y'(x) \) into the differential equation.
3. Simplify to verify the correctness of the solution.

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### c) \( x^2y'' - xy' + y = 0 \)
Proposed solution: \( y(x) = cx - x \ln x \), where \( c \in \mathbb{R} \) and the interval is \( I = (0, \infty) \).

**Steps:**
1. Compute \( y'(x) \) and \( y''(x) \).
2. Substitute \( y(x) \), \( y'(x) \), and \( y''(x) \) into the differential equation.
3. Simplify the terms to check if the equation holds.

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### d) \( y'' - 4y' + 4y + 8\sin 2x = 0 \)
Proposed solution: \( y(x) = c_1 e^{2x} + c_2 x e^{2x} - \cos(2x) \), where \( c_1, c_2 \in \mathbb{R} \) and the interval is \( I = (-\infty, \infty) \).

**Steps:**
1. Find \( y'(x) \) and \( y''(x) \).
2. Plug \( y(x) \), \( y'(x) \), and \( y''(x) \) into the differential equation.
3. Simplify and verify if the equation is satisfied.

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Would you like me to work through the details of any specific part of this problem?

### 5 Related Questions:
1. What is the role of boundary conditions in determining unique solutions for differential equations?
2. How can we solve first-order differential equations like the one in part (b)?
3. What methods are used to solve second-order linear differential equations as in part (d)?
4. What are some common techniques for finding particular solutions to non-homogeneous equations?
5. Why is the interval of validity important in differential equation solutions?

### Tip:
When checking if a function is a solution to a differential equation, always simplify the terms after substitution to see if the equation is satisfied identically.

Zeigen Sie, dass die angegebenen Funktionen Lösungen der jeweiligen Differenzialgleichungen auf den Intervallen I sind.

This page demonstrates how to verify that the provided functions are solutions to their respective differential equations across specific intervals. Explore first and second-order equations, non-homogeneous cases, and key calculus concepts.

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