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The problem involves finding the equivalent ordinary differential equation for the Laplace transform \( U(x, p) \) of a given partial differential equation with initial conditions.

### Given:
- The equation is: \( u_{xx} - u_{tt} = x t \)
- Initial conditions are:
  - \( u(x, 0) = 0 \)
  - \( u_t(x, 0) = 0 \)
  - \( u(0, t) = 0 \)

The goal is to transform this PDE using the Laplace transform in the time domain, where the Laplace transform of \( u(x, t) \) is represented as \( U(x, p) \). 

### Step-by-step process:
1. **Laplace Transform of Time Derivatives**:
   - Taking the Laplace transform of \( u_{tt}(x, t) \), we apply the property:
     \[
     \mathcal{L}\{ u_{tt}(x, t) \} = p^2 U(x, p) - p u(x, 0) - u_t(x, 0)
     \]
     Using the initial conditions \( u(x, 0) = 0 \) and \( u_t(x, 0) = 0 \), we get:
     \[
     \mathcal{L}\{ u_{tt}(x, t) \} = p^2 U(x, p)
     \]

2. **Laplace Transform of the Whole Equation**:
   - Transforming the PDE \( u_{xx} - u_{tt} = xt \), we get:
     \[
     u_{xx} - p^2 U(x, p) = \mathcal{L}\{ x t \}
     \]
   - The Laplace transform of \( x t \) is \( \frac{x}{p^2} \). Thus, the equation becomes:
     \[
     u_{xx} - p^2 U(x, p) = \frac{x}{p^2}
     \]

3. **Rewriting in Terms of \( U(x, p) \)**:
   - We rearrange to find the ordinary differential equation for \( U(x, p) \):
     \[
     u_{xx} = p^2 U(x, p) + \frac{x}{p^2}
     \]

4. **Selecting the Correct Option**:
   - The equation \( u_{xx} = p^2 U(x, p) + \frac{x}{p^2} \) matches the third option in the provided multiple-choice answers:
     \[
     \frac{d^2 U}{dx^2} + p^2 U = \frac{x}{p^2}
     \]

Thus, the correct answer is the third option.

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Do you need further clarification on any step? Here are five related questions:

1. How does the Laplace transform simplify solving PDEs?
2. Why do the initial conditions impact the transformed equation?
3. What is the Laplace transform of higher-order time derivatives?
4. How do boundary conditions like \( u(0, t) = 0 \) affect the solution?
5. How would the solution change if the initial conditions were non-zero?

**Tip:** When applying Laplace transforms to differential equations, remember to account for the initial conditions, as they directly influence the transformed equation.

If Laplace transform of u(x, t) is given by U(x, p), find the equivalent ordinary differential equation for the given initial value problem: u_{xx} - u_{tt} = x t with u(x, 0) = 0, u_t(x, 0) = 0, u(0, t) = 0.

Explore how to solve the partial differential equation u_xx - u_tt = x t using the Laplace transform, with given initial conditions. This guide includes step-by-step transformations and the final ordinary differential equation for U(x, p).

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