https://file.aiquickdraw.com/mathbot/file/1724933635283z0ynj1cy.jpg

The image shows the following integral equation:

\[
\int_{0}^{x} \frac{\phi(t)}{\sqrt{x - t}} \, dt = \sqrt{x}
\]

This is a form of a **Volterra integral equation of the first kind**, where the function \(\phi(t)\) is the unknown function to be determined.

### Solving the Integral Equation

To solve this type of equation, we can use the **Laplace transform** method or use known results for integral equations. The equation given is of a special type where the solution is known to involve the **Dirichlet kernel** or can be found using **Fourier transforms**.

Given that the right-hand side is \(\sqrt{x}\), one known result is:

\[
\phi(t) = \frac{1}{\sqrt{\pi}}
\]

This solution can be verified by substituting \(\phi(t)\) back into the integral equation and evaluating it to ensure that the equation holds true.

Would you like a detailed step-by-step verification or further explanation on how this result is derived? Here are five related questions that might help you deepen your understanding:

1. What is a Volterra integral equation, and how does it differ from Fredholm integral equations?
2. How does the Laplace transform method help in solving integral equations?
3. What are some practical applications of Volterra integral equations in physics and engineering?
4. How can you verify the solution \(\phi(t) = \frac{1}{\sqrt{\pi}}\) for this specific equation?
5. Can this integral equation be solved using numerical methods? If so, how?

**Tip:** When dealing with integral equations, always check if there are known solutions or methods applicable to specific forms of the equation, such as using transforms or kernels.

Learn how to solve the Volterra integral equation of the first kind with a \( \sqrt{x} \) right-hand side. This equation involves the Laplace transform method and Fourier transforms. Explore the solution involving the Dirichlet kernel and verify \( \phi(t) = \frac{1}{\sqrt{\pi}} \). Suitable for advanced mathematics studies.

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