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The problem asks to investigate the **existence and uniqueness** of the solution to the initial value problem:

\[
\frac{dy}{dx} = |y|^{3/4}, \quad y(0) = 0.
\]

This is a first-order differential equation. To analyze the existence and uniqueness, we can apply **Picard-Lindelöf Theorem** (also known as the **existence and uniqueness theorem**), which gives conditions under which a unique solution exists near an initial point.

### Picard-Lindelöf Theorem:
For the initial value problem \( \frac{dy}{dx} = f(x, y) \), \( y(x_0) = y_0 \), there exists a unique solution if:
1. **Continuity**: \( f(x, y) \) is continuous in a neighborhood around \( (x_0, y_0) \).
2. **Lipschitz condition**: \( f(x, y) \) satisfies a Lipschitz condition with respect to \( y \), meaning there is a constant \( L \) such that for all \( y_1 \) and \( y_2 \) in the neighborhood:
   \[
   |f(x, y_1) - f(x, y_2)| \leq L |y_1 - y_2|.
   \]

Now, let's apply these conditions to the given problem.

### Step 1: Continuity of \( f(y) = |y|^{3/4} \)
The function \( f(y) = |y|^{3/4} \) is continuous for all values of \( y \), including at \( y = 0 \). Therefore, the continuity condition is satisfied.

### Step 2: Lipschitz Condition
We now check whether \( f(y) = |y|^{3/4} \) satisfies a Lipschitz condition near \( y = 0 \).

We compute the derivative of \( f(y) = |y|^{3/4} \) to examine its behavior:
- For \( y > 0 \), \( f'(y) = \frac{3}{4} y^{-1/4} \).
- For \( y = 0 \), the derivative becomes problematic because \( y^{-1/4} \to \infty \) as \( y \to 0^+ \).

Thus, \( f(y) = |y|^{3/4} \) does not satisfy a Lipschitz condition at \( y = 0 \), because the derivative is not bounded near \( y = 0 \).

### Conclusion
Since the function \( f(y) = |y|^{3/4} \) does not satisfy the Lipschitz condition near \( y = 0 \), the **Picard-Lindelöf theorem** does not guarantee **uniqueness** of the solution at \( y(0) = 0 \). However, **existence** is still assured due to the continuity of \( f(y) \).

Therefore, the solution exists, but it may not be unique.

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Would you like further details or clarification on any part of the explanation?

Here are 5 questions to explore the topic further:
1. What is the general solution to the differential equation \( \frac{dy}{dx} = |y|^{3/4} \)?
2. How would the behavior of the solution change if \( y(0) \neq 0 \)?
3. Can the Picard-Lindelöf theorem be extended to higher-order differential equations?
4. What happens if the exponent in \( |y|^{3/4} \) is changed? Does uniqueness always fail for such exponents?
5. How can numerical methods be used to approximate solutions in cases where uniqueness fails?

**Tip**: For differential equations where uniqueness might fail, studying the direction fields or using numerical methods like Euler's method can provide insight into the solution's behavior.

Investigate the existence and uniqueness of the solution of the initial value problem: dy/dx = |y|^(3/4), y(0) = 0.

Explore the existence and uniqueness of solutions to the initial value problem dy/dx = |y|^(3/4) with y(0) = 0. This problem examines key concepts in differential equations, including the Picard-Lindelöf theorem, continuity, and the Lipschitz condition, focusing on cases where uniqueness fails due to a non-bounded derivative.

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