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The image you uploaded contains a second-order linear differential equation, and it looks like you are tasked with solving it under the initial condition \( y(1) = 0 \).

The equation given appears to be:
\[
x^2 y'' \cos \frac{y}{x} - y' = x^2 + x y'' \cos \frac{y}{x}
\]

You are also asked to choose between different possible solutions for the equation. The options provided seem to be solutions for the function \( y(x) \). They are as follows:

- **A:** \( y = e^x - 1 \)
- **B:** \( y = \sin^2 \frac{x}{2} \)
- **C:** \( y = e^{-x^2} \)
- **D:** \( y = \sin^2 x + 1 \)

Let me solve or analyze this differential equation to determine which solution satisfies the given equation and initial condition \( y(1) = 0 \).

I'll first attempt to rewrite the equation and find a general solution.The differential equation is:

\[
x^2 \cos \left( \frac{y}{x} \right) y'' - y' = x^2 + x \cos \left( \frac{y}{x} \right) y''
\]

To solve it analytically or numerically and match the given options, we need to explore simplifications or possible direct integrations. Given the complexity of this equation, we should substitute each of the given options into the equation and check which one satisfies it while adhering to the initial condition \( y(1) = 0 \).

Let me now check each option systematically.After simplifying each equation, none of the options immediately solve the differential equation directly in its current form without further assumptions or conditions.

Given that all four options involve complexities in their trigonometric and exponential forms, let's now check which of these satisfies the initial condition \( y(1) = 0 \). We'll evaluate each at \( x = 1 \).

Let me calculate these values for all options.Evaluating the options at \( x = 1 \), we get the following results for \( y(1) \):

- \( y_1(1) = e - 1 \) (which is approximately 1.718, not zero)
- \( y_2(1) = \sin^2(1/2) \) (which is approximately 0.229, not zero)
- \( y_3(1) = e^{-1} \) (which is approximately 0.367, not zero)
- \( y_4(1) = \sin^2(1) + 1 \) (which is approximately 1.708, not zero)

None of these satisfy the condition \( y(1) = 0 \). Therefore, none of the provided options seem to fit both the differential equation and the initial condition. It might suggest the problem could involve further refinement or assumptions about the structure of the solutions.

Do you need further clarification or want to explore another method for solving this? Also, here are some related questions you could consider:

1. How can we simplify complex trigonometric functions in differential equations?
2. Are there alternative methods to solve second-order non-linear differential equations?
3. How do initial conditions affect the behavior of differential equations?
4. What are some typical forms of solutions for second-order linear differential equations?
5. Could numerical methods help in verifying solutions for such complex equations?

**Tip:** Always verify initial conditions after solving a differential equation to ensure the solution is valid for the problem constraints.

Solve the second-order linear differential equation x^2 y'' cos(y/x) - y' = x^2 + x y'' cos(y/x) for y(1) = 0. Choose the correct solution from the options.

This problem involves solving a second-order linear differential equation with initial condition y(1) = 0. The equation requires knowledge of differential equations and trigonometric functions. Learn how to approach and verify the solution.

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