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The problem you've uploaded involves solving a second-order differential equation along with initial conditions to determine \( y\left(\frac{\pi}{2}\right) \).

Let's break down the problem from what can be seen:

### Given Differential Equation:
\[
x^2 y'' + x y' + \left(x - \frac{1}{4}\right)y = \sin(x), \quad x > 0
\]
### Initial Conditions:
\[
y(0) = 0, \quad y'(0) = 1
\]

The task is to find \( y\left(\frac{\pi}{2}\right) \).

### Approach to Solving the Problem

1. **Recognizing the Equation Type**:
   - This is a second-order linear non-homogeneous differential equation with variable coefficients.

2. **Using Series Solutions or Standard Methods**:
   - One typical approach would involve solving the homogeneous part first, followed by finding a particular solution for the non-homogeneous part.

3. **Numerical or Exact Solution**:
   - Given the complexity of the equation, an analytical solution may not be straightforward, and numerical methods (like using the Runge-Kutta method or similar) might be more effective to compute \( y\left(\frac{\pi}{2}\right) \).

### Solution Strategy:

I will set up the necessary steps to solve this equation numerically or using standard mathematical techniques and calculate \( y\left(\frac{\pi}{2}\right) \). Let's proceed with solving it.

I'll now compute it numerically.The numerical solution for \( y\left(\frac{\pi}{2}\right) \) is approximately:

\[
y\left(\frac{\pi}{2}\right) \approx 18.16
\]

Would you like more details on the steps or methods used, or do you have any other questions?

Here are 5 related questions you might consider:
1. How does the method of solving differential equations numerically work?
2. What are the advantages and disadvantages of using numerical methods for solving differential equations?
3. Can this differential equation be solved using series solutions?
4. How do initial conditions affect the behavior of solutions in differential equations?
5. What is the Runge-Kutta method and how is it applied in solving differential equations?

**Tip**: When solving complex differential equations, consider whether an analytical solution is possible or if numerical methods (like Runge-Kutta) are more practical.

Find y(π/2) for the differential equation: x^2 y'' + x y' + (x - 1/4)y = sin(x), with initial conditions y(0) = 0 and y'(0) = 1.

Solve the second-order differential equation x^2 y'' + x y' + (x - 1/4)y = sin(x) with initial conditions y(0) = 0 and y'(0) = 1. Learn how to use numerical methods such as the Runge-Kutta method to approximate y(π/2). Suitable for college-level students.

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