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The image contains a problem about differential equations and asks for the correct statement among the four options provided. Here's a transcription of the question and the options:

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**Q7: On considère les équations différentielles suivantes :**

\[
(E_1) : x y'' + 2y' - xy = 2e^x \quad \text{et} \quad (E_2) : z'' - z = 2e^x.
\]

Parmi les affirmations suivantes, cocher celle qui est vraie :

A. Si \( y \) est une solution de \((E_1)\), alors \( z = xy \) est une solution de \((E_2)\).

B. \((E_1)\) admet une solution particulière sous la forme \( y_0 = ae^x \), \( a \in \mathbb{R} \).

C. La solution générale de \((E_1)\) sur \( ]0, +\infty[ \) est \( y = \left( \frac{a}{x} + 1 \right) e^x + \left( \frac{b}{x} + 1 \right) e^{-x} \), \( a, b \in \mathbb{R} \).

D. \((E_1)\) n'admet pas de solution sur \( \mathbb{R} \).

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### Analyzing the statements:

1. **Statement A**: It suggests that if \( y \) is a solution of \( (E_1) \), then \( z = xy \) is a solution of \( (E_2) \). We need to verify whether substituting \( z = xy \) into \( (E_2) \) produces a valid solution.
   
2. **Statement B**: It claims that there exists a particular solution of \( (E_1) \) of the form \( y_0 = ae^x \). We can verify this by substituting \( y_0 = ae^x \) into \( (E_1) \) to check if it satisfies the equation.

3. **Statement C**: This presents the general solution of \( (E_1) \) for \( x \in ]0, +\infty[ \) as a combination of exponential terms. To check this, we must solve the differential equation \( (E_1) \) and compare the result with the given expression.

4. **Statement D**: It states that \( (E_1) \) has no solution over \( \mathbb{R} \), which implies that the equation is unsolvable in \( \mathbb{R} \). This can be tested by analyzing the nature of \( (E_1) \).

### Would you like me to go through the verification of each statement in detail?

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Here are 5 related questions to expand on the topic:

1. What methods can be used to solve second-order non-homogeneous differential equations like \( (E_1) \) and \( (E_2) \)?
2. How can we apply the method of undetermined coefficients to find particular solutions of non-homogeneous differential equations?
3. What are the conditions for transforming a solution of one differential equation into a solution of another related equation?
4. How can the general solution of a second-order differential equation be constructed from the homogeneous solution and a particular solution?
5. What is the significance of boundary conditions in determining the unique solution to a differential equation?

**Tip**: When solving second-order differential equations, always solve the homogeneous part first to find the complementary solution, then find a particular solution to the non-homogeneous equation.

On considère les équations différentielles suivantes : (E1) : x y'' + 2y' - xy = 2e^x et (E2) : z'' - z = 2e^x. Parmi les affirmations suivantes, cocher celle qui est vraie : A. Si y est une solution de (E1), alors z = xy est une solution de (E2). B. (E1) admet une solution particulière sous la forme y0 = ae^x, a ∈ R. C. La solution générale de (E1) sur ]0, +∞[ est y = (a/x + 1) e^x + (b/x + 1) e^(-x), a, b ∈ R. D. (E1) n'admet pas de solution sur R.

Explore the solution of second-order differential equations such as (E1): x y'' + 2y' - xy = 2e^x and (E2): z'' - z = 2e^x. Learn how to verify the correctness of statements about particular and general solutions in the context of differential equations.

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